Automorphism Groups of Maps, Surfaces and Smarandache Geometries (second edition), graduate text book in mathematics, by Linfan Mao

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Graduate Textbook in Mathematics

LINFAN MAO

AUTOMORPHISM GROUPS OF MAPS, SURFACES AND

SMARANDACHE GEOMETRIES

Second Edition

A

O

B C

The Education Publisher Inc.

2011



Linfan MAO

Academy of Mathematics and Systems

Chinese Academy of Sciences

Beijing 100190, P.R.China

and

Beijing Institute of Civil Engineering and Architecture

Beijing 100044, P.R.China

Email: [email protected]

Automorphism Groups of Maps, Surfaces and

Smarandache Geometries

Second Edition

The Education Publisher Inc.

2011



This book can be ordered from:

The Educational Publisher, Inc.

1313 Chesapeake Ave.Columbus, Ohio 43212, USA

Toll Free: 1-866-880-5373

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Website: www.EduPublisher.com

Peer Reviewers:

F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing

100080, P.R.China.

Y.P.Liu and R.X.Hao, Department of Applied Mathematics, Beijing Jiaotong University,

Beijing 100044, P.R.China. S.Bhattacharya, Alaska Pacific University, AK, USA.

H.Iseri, Department of Mathematics and Computer Information Science, Mansfield Uni-

versity, Mansfield, PA 16933, USA.

Copyright 2011 by The Education Publisher Inc. and Linfan Mao

Many books can be downloaded from the following Digital Library of Science:

http: // www.gallup.unm.edu / ∼smarandache / eBooks-otherformats.htm

ISBN: 978-1-59973-154-4

Printed in America



Preface to the Second Edition

Automorphism groups survey similarities on mathematical systems, which appear nearly

in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and

theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high

symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally,

automorphism groups enable one to distinguish systems by similarity. More automor-

phisms imply more symmetries of that system. This fact has established the fundamental

role of automorphism groups in modern sciences. So it is important for graduate students

knowing automorphism groups with applications.

The first edition of this book is in fact consisting of my post-doctoral reports in Chi-

nese Academy of Sciences in 2005, not self-contained and not suitable as a textbook for

graduate students. Many friends of mine suggested me to extend it to a textbook for

graduate students in past years. That is the initial motivation of this edition. Besides, I

also wish to survey applications of Smarandache’s notion with combinatorics, i.e., math-

ematical combinatorics to automorphism groups of maps, surfaces and Smarandache ge-

ometries in this edition. The two objectives advance me to complete this self-contained

book.

Indeed, there are many ways for introducing automorphism groups. I plan them for

graduate students both in combinatorics and geometry. The materials in this book includegroups with actions, graphs with symmetries, graphs on surfaces with enumeration, reg-

ular maps, isometries on finitely or infinitely pseudo-Euclidean spaces and an interesting

notion for developing mathematical sciences in 21th century, i.e. the CC conjecture.

Contents in in this book are outlined following.

Chapters 1 and 2 are an introduction to groups. Topics such as those of groups and



ii Automorphism Groups of Maps, Surfaces and Smarandache Geometries

subgroups, regular representations, homomorphism theorems, structures of finite Abelian

groups, transitive groups, automorphisms of groups, characteristic subgroups, p-groups,

primitive groups, regular normal subgroups are discussed and a few useful results, for ex-

amples, these Burnside lemma, Sylow theorem and O’Nan-Scott theorem are established.Furthermore, an elementary introduction to multigroups and permutation multigroups, in-

cluding locally or globally transitive groups, locally or globally regular groups can be also

found in Chapters 1 and 2.

For getting automorphism groups of graphs, these symmetric graphs, including vertex-

transitive graphs, edge-transitive graphs, arc-transitive graphs and semi-arc transitive graphs

are introduced in Chapter 3. Indeed, the automorphism group of a normally Cayley graph

or GRR of a finite group can be completely determined. For classifying maps on sur-

faces underlying a graph G, one needs to consider the action of semi-arc automorphismgroup Aut 1

2G on its semi-arc set X 1

2G. Such groups are not very diff erent from that of

automorphism group of G. In fact, Aut 12

G = AutG if G is loop-free. This chapter also

discuses multigroup action graphs, which make a few results on globally transitive groups

in Chapter 2 simple.

As a preparing for combinatorial maps with applications to Klein surfaces, Chapter

4 is mainly on surfaces, including both topological surfaces and Klein surfaces. Indeed,

Sections 4.1-4.3 can be used to an introduction on topological surfaces and Sections 4.4-

4.5 on Klein surfaces. These fundamental techniques or results on surfaces, such as thoseof classifying theorem of surfaces by elementary operations, Seifert-Van Kampen theo-

rem, fundamental groups of surfaces, NEC groups and automorphism groups of Klein

surfaces are well discussed in this chapter.

Chapters 5-7 are an introduction on algebraic maps, i.e., graphs on surfaces, partic-

ularly, automorphisms of maps. The rotation embedding scheme on graphs and its con-

tribution to algebraic maps can be found in Sections 5.1-5.2. Then map groups, regular

maps and the technique for constructing regular maps by triangle groups are interpreted

in Sections 5.3-5.5.

Chapter 6 concentrates on lifting automorphisms of maps by that of voltage assign-

ment technique. A condition for a group being that of a lifted map and a combinatorial

refinement of the Hurwitz theorem on Riemann surfaces are gotten in Sections 6 .1-6.4.

After that, Section 6.5 concerns the order of an automorphism of Klein surfaces by that

of map, which is an interesting problem in Klein surfaces.



Preface to the Second Edition iii

The objective of Chapter 7 is to find presentations of automorphisms of maps un-

derlying a graph. A general condition for a graph group being that of map is established

in the first section. Then all these presentations for automorphisms of maps underlying a

complete graph, a semi-regular graph or a bouquet are found, which are useful for enu-merating maps underlying such a graph.

Applying results in Chapter 7 enables one to classify maps, i.e., enumerating rooted

maps or maps underlying a graph in Chapter 8. These enumerating results on rooted

maps underlying a graph are presented in Sections 8.1-8.2 by group action. It is worth

to celebrate that a sum-free formula for rooted maps underlying a graph is found by the

action semi-arc automorphism group of graph. Then a general scheme for enumerating

maps underlying a graph is established in Section 8.3. By applying this scheme and those

presentations of automorphisms of maps in Chapter 7, these complete maps, semi-regularmaps and one-vertex maps are enumerated in Sections 8 .4-8.6, respectively.

Chapter 9 turns on a special kind of automorphisms, i.e., isometries on Smarandache

geometry, a mixed geometry with an axiom validated or invalided, or only invalided but in

at least two distinct ways. A formally definition with examples for such geometry can be

found in Sections 9.1-9.2. Then all isometries on finitely or infinitely pseudo-Euclidean

spaces (Rn, µ) are determined in Sections 9.3-9.4. It should be noted that for the finite

case, all such isometries can be combinatorially characterized by graphs embedded in the

Euclidean space Rn.

The final chapter concentrates on an important notion for developing mathematical

sciences in 21th century, i.e., the CC conjecture appeared in Chapter 5 of the first edition

in 2005. That is the originality of mathematical combinatorics. Its contributions to math-

ematics and physics are introduced, and research problems are presented in this chapter.

These interested readers are referred to [Mao25] for its further applications to geometry

or Riemann geometry.

This edition was began to prepare in 2009. Many colleagues and friends of mine

have given me enthusiastic support and endless helps in writing. Here I must mention

some of them. On the first, I would like to give my sincerely thanks to Dr.Perze for his

encourage and endless help. Without his encourage, I would do some else works, can not

investigate mathematical combinatorics for years and finish this edition. Second, I would

like to thank Professors Feng Tian, Yanpei Liu, Mingyao Xu, Jiyi Yan, Fuji Zhang and

Wenpeng Zhang for them interested in my research works. Their encouraging and warm-



iv Automorphism Groups of Maps, Surfaces and Smarandache Geometries

hearted supports advance this book. Thanks are also given to Professors Han Ren, Yanqiu

Huang, Junliang Cai, Rongxia Hao, Wenguang Zai, Goudong Liu, Weili He and Erling

Wei for their kindly helps and often discussing problems in mathematics altogether. Par-

tially research results of mine were reported at Chinese Academy of Mathematics & Sys-tem Sciences, Beijing Jiaotong University, Beijing Normal university, East-China Normal

University and Hunan Normal University in past years. Some of them were also reported

at The 2nd and 3rd Conference on Graph Theory and Combinatorics of China in 2006

and 2008. My sincerely thanks are also give to these audiences discussing mathematical

topics with me in these periods.

Of course, I am responsible for the correctness all of these materials presented here.

Any suggestions for improving this book or solutions for open problems in this book are

welcome.

L.F.Mao

June 24, 2011



Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Chapter 1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

§1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 . 1 . 2 C a r d i n a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Subset Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

§1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Algebra System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Group Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.5 Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.6 Regular Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

§1.3 Homomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.2 Quotient Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Isomorphism Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

§1.4 Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 Direct Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.2 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4.3 Finite Abelian Group Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

§1.5 MultiGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5.1 MultiGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1 . 5 . 2 S u b m u l t i g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

1 . 5 . 3 N o r m a l S u b m u l t i g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

1.5.4 Abelian Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5.5 Bigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.5.6 Constructing Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

§1.6 R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9



vi Automorphism Groups of Maps, Surfaces and Smarandache Geometries

Chapter 2 Action Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

§2.1 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.1 Group Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.2 Stabilizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1.3 Burnside Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

§2.2 Transitive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.1 Transitive Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 . 2 . 2 M u l t i p l y T r a n s i t i v e G r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6

2.2.3 Sharply k-Transitive Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

§2.3 Automorphisms of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.1 Automorphism Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3.2 Characteristic Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.3 Commutator Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

§2.4 P-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4.1 Sylow Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4.2 Application of Sylow Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.4.3 Listing p-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

§2.5 Primitive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5.1 Imprimitive Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5.2 Primitive Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5.3 Regular Normal Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.5.4 O Nan-Scott Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

§2.6 Local Action and Extended Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.6.1 Local Action Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.6.2 Action Extended Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.6.3 Action MultiGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

§2.7 R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7

Chapter 3 Graph Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9

§3.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.1.1 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.1.2 Graph Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.3 Graph Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85



Contents vii

3.1.4 Smarandachely Graph Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9

§3.2 Graph Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2.1 Graph Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2.2 Graph Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.2.3 Γ- A c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4

§3.3 Symmetric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.3.1 Vertex-Transitive Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.3.2 Edge-Transitive Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.3.3 Arc-Transitive Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

§3.4 Graph Semi-Arc Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.4.1 Semi-Arc Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.4.2 Graph Semi-Arc Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.4.3 Semi-Arc Transitive Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

§3.5 Graph Multigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5.1 Graph Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5.2 Multigroup Action Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5.3 Globally Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

§3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 4 Surface Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

§4.1 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1.1 Topological Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1.2 Continuous Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.1.3 Homeomorphic Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.1.4 Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.1.5 Quotient Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

§4.2 Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2.1 Connected Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.2 Polygonal Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4 . 2 . 3 E l e m e n t a r y E q u i v a l e n c e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 8

4.2.4 Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4 . 2 . 5 E u l e r C h a r a c t e r i s t i c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4

§4.3 Fundamental Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136



viii Automorphism Groups of Maps, Surfaces and Smarandache Geometries

4.3.1 Homotopic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

4 . 3 . 2 F u n d a m e n t a l G r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 7

4.3.3 Seifert-Van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.3.4 Fundamental Group of Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

§4.4 NEC Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.4.1 Dianalytic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.4.2 Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.4.3 Morphism of Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.4.4 Planar Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.4.5 NEC Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

§4.5 Automorphisms of Klien Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.5.1 Morphism Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.5.2 Double Covering of Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.5.3 Discontinuous Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.5.4 Automorphism of Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

§4.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Chapter 5 Map Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 7

§5.1 Graphs on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.1.1 Cell Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.1.2 Rotation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.1.3 Equivalent Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 72

5.1.4 Euler-Poincare Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

§5.2 Combinatorial Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.2.1 Combinatorial Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.2.2 Dual Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.2.3 Orientability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.2.4 Standard Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179§5.3 Map Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5 . 3 . 1 I s o m o r p h i s m o f M a p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 1

5 . 3 . 2 A u t o m o r p h i s m o f M a p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 4

5.3.3 Combinatorial Model of Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

§5.4 Regular Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191



Contents ix

5.4.1 Regular Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

5.4.2 Map NEC-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

5.4.3 Cayley Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.4.4 Complete Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197§5.5 Constructing Regular Maps by Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.5.1 Regular Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.5.2 Regular Map on Finite Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

5.5.3 Regular Map on Finite Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

§5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Chapter 6 Lifting Map Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

§6.1 Voltage Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.1.1 Covering Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.1.2 Covering Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.1.3 Voltage Map with Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

§6.2 Group being That of a Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.2.1 Lifting Map Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.2.2 Map Exponent Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.2.3 Group being That of a Lifted Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

§6.3 Measures on Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.3.1 Angle on Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.3.2 Non-Euclid Area on Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

§6.4 A Combinatorial Refinement of Huriwtz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.4.1 Combinatorially Huriwtz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.4.2 Application to Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

§6.5 The Order of Automorphism of Klein Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.5.1 The Minimum Genus of a Fixed-Free Automorphism . . . . . . . . . . . . . . . . . . 233

6.5.2 The Maximum Order of Automorphisms of a Map . . . . . . . . . . . . . . . . . . . . . 236

§6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Chapter 7 Map Automorphisms Underlying a Graph . . . . . . . . . . . . . . . . . . . . . . . . . 241

§7.1 A Condition for Graph Group Being That of Map. . . . . . . . . . . . . . . . . . . . . . . . . . . 242

7.1.1 Orientation-Preserving or Reversing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

7.1.2 Group of a Graph Being That of Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 44



x Automorphism Groups of Maps, Surfaces and Smarandache Geometries

§7.2 Automorphisms of a Complete Graph on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

7.2.1 Complete Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

7.2.2 Automorphisms of Complete Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

§7.3 Map-Automorphism Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2567.3.1 Semi-Regular Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

7 . 3 . 2 M a p - A u t o m o r p h i s m G r a p h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 0

§7.4 The Automorphisms of One-Face Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.4.1 One-Face Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.4.2 Automorphisms of One-Face Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

§7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Chapter 8 Enumerating Maps on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

§8.1 Roots Distribution on Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.1.1 Roots on Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.1.2 Root Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

8.1.3 Rooted Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

§8.2 Rooted Map on Genus Underlying a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.2.1 Rooted Map Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8.2.2 Rooted Map Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

§8.3 A Scheme for Enumerating Maps Underlying a Graph. . . . . . . . . . . . . . . . . . . . . . .283

§8.4 The Enumeration of Complete Maps on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

§8.5 The Enumeration of Maps Underlying a Semi-Regular Graph . . . . . . . . . . . . . . . . 292

8.5.1 Crosscap Map Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.5.2 Enumerating Semi-Regular Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

§8.6 The Enumeration of a Bouquet on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

8.6.1 Cycle Index of Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

8.6.2 Enumerating One-Vertex Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

§8.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Chapter 9 Isometries on Smarandache Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 6

§9.1 Smarandache Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.1.1 Geometrical Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.1.2 Smarandache Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.1.3 Mixed Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308



Contents xi

§9.2 Classifying Iseri s Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

9.2.1 Iseri s Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

9.2.2 A Model of Closed Iseri s Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

9.2.3 Classifying Closed Iseri

s Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311§9.3 Isometries of Smarandache 2-Manfiolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

9.3.1 Smarandachely Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

9.3.2 Isometry on R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

9.3.3 Finitely Smarandache 2-Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

9.3.4 Smarandachely Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

9.3.5 I nfinitely Smar andache 2- Manif old. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327

§9.4 Isometries of Pseudo-Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

9.4.1 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3329.4.2 Linear Isometry on Euclidean Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .333

9.4.3 Isometry on Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

9.4.4 Pseudo-Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

9.4.5 Isometry on Pseudo-Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

§9.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Chapter 10 CC Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

§10.1 CC Conjecture on Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

1 0 . 1 . 1 C o m b i n a t o r i a l S p e c u l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 7

10.1.2 CC Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

10.1.3 CC Problems in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

§10.2 CC Conjecture to Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

1 0 . 2 . 1 C o n t r i b u t i o n t o A l g e b r a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 2

10.2.2 Contribution to Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

§10.3 CC Conjecture to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

10.3.1 M-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10.3.2 Combinatorial Cosmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377



xii Automorphism Groups of Maps, Surfaces and Smarandache Geometries

There are many wonderful things in nature, but the most wonderful

of all is man.

Sophocles, an ancient Greek dramatist



CHAPTER 1.

Groups

A group is surely the laws of combinations on its symbols, an important con-ception of mathematics. One classifies groups into two categories, i.e., the ab-

stract groups and permutation groups. Its application fields includes physics,

chemistry, biology, crystallography,..., etc.. Now it has become a fundamental

of all branches of mathematical sciences. For introducing readers to abstract

groups, these algebraic systems, groups with subgroups, regular representa-

tion, homomorphism theorems, Abelian groups with structures, multigroups

and submultigroups with elementary properties are discussed in this chapter,

where multigroups are generalized algebraic systems of groups by Smaran-

dache multi-space, i.e., a union of groups, diff erent two by two.



2 Chap.1 Groups

§1.1 SETS

1.1.1 Set. A set S is a category consisting of parts, i.e., a collection of objects possessing

with a property P. Usually, a set S is denoted by

S = x | x possesses the property P .

If an element x possesses the property P, we say that x is an element of the setS, denoted

by x ∈ S. On the other hand, if an element y does not possesses the property P , then it

is not an element of S, denoted by y S.

For examples,

Z + =

1, 2,

· · ·, n,

· · · ,

P = cities with more than 2 million peoples in China,

E = ( x, y)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1

are 3 sets by definition, and the number n ≥ 1, city with more than 2 million peoples in

China and point ( x, y) with 0 ≤ x, y ≤ 1 are elements of sets Z +, P and E , respectively.

Let S , T be two sets. These binary operations union S ∪ T and intersection S ∩ T of

sets S and T are defined by

S T = x| x ∈ S or x ∈ T , S T = x| x ∈ S and x ∈ T .

These operations ∪ and ∩ have the following laws.

Theorem 1.1.1 Let X , T and R be sets. Then

(i) X

X = X a n d X

X = X;

(ii) X

T = T

X and X

T = T

X;

(iii) X (T R) = ( X T ) R and X (T R) = ( X T ) R;

(iv) X (T R) = ( X T )( X R) ,

X

(T

R) = ( X

T )

( X

R).

Proof These laws (i)-(iii) can be verified immediately by definition. For the law (iv),

let x ∈ X

(T

R) = ( X

T )

( X

R). Then x ∈ X or x ∈ T

R, i.e., x ∈ T and

x ∈ R. Now if x ∈ X , we know that x ∈ X ∪ T and x ∈ X ∪ R. Whence, we get that



Sec.1.1 Sets 3

x ∈ ( X

T )

( X

R). Otherwise, x ∈ T

R, i.e., x ∈ T and x ∈ R. We also get that

x ∈ ( X

T )

( X

R).

Conversely, for ∀ x ∈ ( X

T )

( X

R), we know that x ∈ X

T and x ∈ X

R,

i.e., x ∈ X or x ∈ T and x ∈ R. If x ∈ X , we get that x ∈ X (T R). If x ∈ T and x ∈ R, we also get that x ∈ X

(T

R). Therefore, X

(T

R) = ( X

T )

( X

R) by

definition.

Similarly, we can also get the law X ∩ T = X ∪ T .

Let S1 and S2 be two sets. If for ∀ x ∈ S1, there must be x ∈ S2, then we say that

S1 is a subset of S2, denoted by S1 ⊆ S2. A subset S1 of S2 is proper , denoted by

S1 ⊂ S2 if there exists an element y ∈ S2 with y S1 hold. It should be noted that the

void (empty) set ∅ is a subset of all sets by definition. All subsets of a set S naturally

form a set P(S), called the power set of S.

Now let S be a set and X ∈ P(S). We define the complement X of X ⊂ S to be

X = y | y ∈ S but y X .

Then we know the following result.

Theorem 1.1.2 Let S be a set, S , T ⊂ S. Then

(i) X ∩

X =∅

and X ∪

X = S;

(ii) X = X ;

(iii) X ∪ T = X ∩ T and X ∩ T = X ∪ T .

Proof The laws (i) and (ii) can be immediately verified by definition. For (iii), let

x ∈ X ∪ T . Then x ∈ S but x X ∪ T , i.e., x X and x T . Whence, x ∈ X and x ∈ T .

Therefore, x ∈ X ∩ T . Now for ∀ x ∈ X ∩ T , there must be x ∈ X and x ∈ T , i.e., x ∈ Sbut x X and x T . Hence, x X ∪ T . This fact implies that x ∈ X ∪ T . By definition,

we find that X ∪

T = X ∩

T . Similarly, we can also get the law X ∩

T = X ∪

T . This

completes the proof.

1.1.2 Cardinality. A mapping f from a set S to T is a subset of S × T such that for

∀ x ∈ S , | f (∩( x × T )| = 1, i.e., f ∩ ( x × T ) only has one element. Usually, we denote

a mapping f from S to T by f : S → T and f ( x) the second component of the unique

element of f ∩ ( x × T ), called the image of x under f .



4 Chap.1 Groups

A mapping f : S → T is called injection if for ∀ y ∈ T , | f ∩ (S × y)| ≤ 1 and

surjection if | f ∩ (S × y)| ≥ 1. If it is both injection and surjection, i.e., | f ∩ (S × y)| = 1,

then it is called a bijection or a 1 − 1 mapping.

Definition 1.1.1 Let S , T be two sets. If there is a bijection f : S → T, then the

cardinality of S is equal to that of T . Particularly, if T = 1, 2, · · · , n , the cardinal

number, usually called the order of S is defined to be n, denoted by |S | = n.

Definition 1.1.2 A set S is finite if and only if c(S ) < ∞. Otherwise, S is infinite.

Definition 1.1.3 A set S is countable if there is a bijection f : S → Z +.

By this definition, one can enumerate all elements of S by an infinite sequence

s1, s2,

· · ·, sn,

· · ·. These Z +, P and E in Subsection 1.1.1 are countable, finite and infi-

nite set, respectively. Generally, we have the following result.

Theorem 1.1.3 A set S is infinite if and only if it contains a countable subset.

Proof If S contains a countable subset, by Definition 1.1.3 it is infinite. Now if S is

infinite, choose s1 ∈ S , s2 ∈ S \ s1, s3 ∈ S \ s1, s2, ..., sn ∈ S \ s1, s2, · · · , sn−1,.... By

assumption, S is infinite, so for any integer n ≥ 1, the set S \ s1, s2, · · · , sn−1 can never

be empty. Therefore, we can always choose an element sn from it and this process will

never stop until we get an infinite sequence s1, s2, · · · , sn, · · ·, a countable subset of S .

Theorem 1.1.4 The set R of all real numbers is not countable.

Proof Assume there is an enumeration r 1, r 2, · · · , r n, · · · of all real numbers. Then list

the decimal expansion of these numbers after the decimal point in their enumerated order

in a square array:

r 1 = · · · .a11a12a13a14 · · ·r 2 = · · · .a21a22a23a24 · · ·r 3 = · · · .a31a32a33a34 · · ·

r 4 = · · · .a41a42a43a44 · · ·· · · · · · · · · · · · · · · · · · · · · · · · ,

where amn is the nth digit after the decimal point of r m. Then we construct a new real

number ζ between 0 and 1 as follows:

Let the bth digit bn in the decimal expansion of b be ann − 1 if ann 0 and 1 if

ann = 0. Then b = .b1b2b3b4 · · · is the decimal expansion of b, which is a real number by



Sec.1.2 Groups 5

definition but diff ers from the nth number r n of the enumeration in the nth decimal place

for any integer n ≥ 1. Whence, b is not in the sequence r 1, r 2, · · · , r n, · · ·. This contradicts

our assumption.

1.1.3 Subset Enumeration. LetS be a countable set, i.e.,

S = s1, s2, · · · , sn, · · · .

We adopt the following convention for subsets.

Convention 1.1.1 For a subset S = si1, si2

, · · · , sil of S , l ≥ 1 , assign it to a monomial

si1si2

· · · sil.

Applying this convention, we can find the generator of subsets of a set S.

Theorem 1.1.5 Under Convention 1.1.1 , the generator of elements in the power set P(S)

is

G(P(S)) =

ǫ s=0 or 1

s∈s

sǫ s .

Proof Let T = si1, si2

, · · · , sl, l ≥ 1 be an element in P(S). Then it is the term

si1si2

· · · sl in G(P(S)). Conversely, let si1si2

· · · sk , k ≥ 1 be a term in G(P(S)). Then it

is the subset si1, si2

, · · · , sk by Convention 1.1.1.

For a finite set S, we can get a closed formula for counting its subsets following.

Theorem 1.1.6 Let S be a finite set. Then the number of its subsets is

|P(S)| = 2|S|.

Proof Notice that for any integer i, 1 ≤ i ≤ |S|, there are

|S|i

subsets of cardinal-

ity i inS. Therefore, we find that

|P(S)| =

|S|i=1

|S|i

= 2|S|.



6 Chap.1 Groups

§1.2 GROUPS

1.2.1 Algebra System. Let A be a nonempty set. A binary operation on A is a bijection

o : A ×

A →

A . Thus o associates each ordered pair (a, b) of elements of A with an

element o(a, b) that of A . For simplicity, we write a b for o(a, b) and refer to as a

binary operation on A . A set A associated with a binary operation is called to be an

algebraic system, denoted by (A ; ).

If A is finite, let A = x1, x2, · · · , xn, we can present an algebraic system (A ; )

easily by operation table following.

x1 x2 · · · xn

x1 x1 x1 x1 x2 · · · x1 xn

x2 x2 x1 x2 x2 · · · x2 xn

· · · · · · · · · · · · · · · xn xn x1 xn x2 · · · xn xn

Table 1.2.1

For example, let K = 1, α , β , γ with an operation determined by the following

table.

1 α β γ

1 1 α β γ

α α 1 γ β

β β γ 1 α

γ γ β α 1

Table 1.2.2

Then we easily get that

1

1 = α

α = β

β = γ

γ = 1,

1 α = α 1 = α, 1 β = β 1 = β, 1 γ = γ 1 = γ ,

α β = β α = γ , α γ = γ α = β, β γ = γ β = α

by Table 1.2.2. Notice that x ( y z) = ( x y) z and x y = y x for ∀ x, y, z ∈ K in Table

1.2.2. These properties enables us to introduce the associative and commutative laws for

operation following.



Sec.1.2 Groups 7

Definition 1.2.1 An algebraic system (A ; ) is associative if for ∀a, b, c ∈ A ,

(a b) c = a (b c).

An associative system (A ; ) is usually called a semigroup. A system (A ; ) is Abelian if

for ∀a, b ∈ A ,

a b = b a.

There are many non-Abelian systems. For example, let M n(R) be all n × n matrixes

with matrix multiplication . We have known that the equality

A B = B A

does not always hold for ∀ A, B ∈ M n(R) from linear algebra. Whence, ( M n(R), )isanon-Abelian system. Notice that each element associated with the element 1 n×n is unchanging

in M n(R). Such an element is called to be a unit defined following, which also enables us

to introduce the inverse element of an element in (A , ).

Definition 1.2.2 Let (A ; ) be an algebraic system. An element 1l ∈ A (or 1r ∈ A , or

1 ∈ A ) is called to be a left unit (or right unit, or unit) if for ∀a ∈ A

1l a = a (or a 1r = a, or 1 a = a 1 = a).

Definition 1.2.3 Let (A ; ) be an algebraic system with a unit 1A . An element b ∈ A is

called to be a right inverse of a ∈ A if a b = 1A .

Certainly, there are algebra systems without unit. For example, let H = a, b, c, d with an operation · determined by the following table.

· a b c d

a b c a d

b c d b ac a b d c

d d a c b

Table 1.2.3

Then ( H , ·) is an algebraic system without unit.



8 Chap.1 Groups

1.2.2 Group. A group is an algebraic associative system with unit and inverse elements,

formally defined in the following.

Definition 1.2.4 An algebraic system (G ;

) is a group if conditions (1)-(3) following

hold:

(1) ( x y) z = x ( y z) , ∀ x, y, z ∈ G ;

(2) ∃1G ∈ G such that 1G x = x 1G = x, x ∈ G ;

(3) ∀ x ∈ G , ∃ y ∈ G such that x y = y x = 1G .

A group (G ; ) is Abelian if it is itself Abelian, i.e., an additional condition (4) fol-

lowing holds:

(4)

∀ x, y

∈G, x

y = y

x,

∀ x, y

∈G .

For example, the system (K ; ) determined by Table 1.2.2 is such an Abelian group,

usually called Klein 4-group. More examples of groups are shown following.

Example 1.2.1(Groups of Numbers) Let Z,Q,R and C denote respectively sets of all

integers, rational numbers, real numbers and complex numbers and +, · the ordinary

addition, multiplication. Then we know

(1) (Z; +), (Q; +), (R; +) and (C; +) are four Abelian infinite groups with identity 0

and inverse

− x for

∀ x

∈Z,Q,R or C;

(2) (Z \ 0; ·), (Q \ 0; ·), (R \ 0; ·) and (C \ 0; ·) are four Abelian infinite groups

with identity 1 and inverse 1/ x for ∀ x ∈ Z,Q,R or C.

(3) Let n be an integer. Define an equivalent relation ∼ on Z following:

a ∼ b ⇔ a ≡ b(modn).

Denoted by i the equivalent class including i. We get n equivalent classes 0, 1, · · · , n − 1.

Let Zn = 0, 1, · · · , n − 1. Then (Zn; +) is an Abelian n-group with identity 0, inverse

− x for x

∈Zn and (Zn

\ 0;·) an Abelian (n

−1)-group with identity 1, inverse 1/ x for

x ∈ Zn \ 0, where 1/ x denotes the equivalent class including such 1/ x with x · (1/ x) ≡1(modn).

Example 1.2.2(Groups of Matrixes) Let GL(n,R) be the set of all invertible n×n matrixes

with coefficients in R and +, · the ordinary matrix addition and multiplication. Then

(1) (GL(n,R); +) is an Abelian infinite group with identity 0n×n, the n ×n zero matrix



Sec.1.2 Groups 9

and inverse − A for A ∈ GL(n,R), where − A is the matrix replacing each entry a by −a in

matrix A.

(2) (GL(n,R); ·) is a non-Abelian infinite group if n ≥ 2 with identity 1n×n, the n × n

unit matrix and inverse A−1 for A ∈ GL(n,R), where A · A−1 = 1n×n. For its non-Abelian,

let n = 2 for simplicity and

A =

1 2

2 1

, B =

2 −3

3 1

.

Calculations show that

1 2

2 1 · 2 −3

3 1 = 8 −1

7 −5 , 2 −3

3 1 · 1 2

2 1 = −4 1

5 7 .

Whence, A · B B · A.

Example 1.2.3(Groups of Linear Transformation) Let V be an n-dimensional vector

space over R and GL(V ,R) the set of all bijection linear transformation of V . We have

known that each bijection linear transformation of V is associated with a non-singular

n ×n matrix and the composition of two such transformations is correspondent with that

of matrixes if a fixed basis of V is chosen. Therefore, (GL(V ,R); ) is a group by Example

1.2.2.

Example 1.2.4(Isometries of E 2) Let E 2 be a Euclidean plane. There are three basic

isometries in E 2, i.e., rotations about a point, reflections in a line and translations moving

a point ( x, y) to ( xa, y + b) for some fixed a, b ∈ R. We have know that any isometry is a

rotation, a reflection, a translation, or their product.

If X is a bounded subset of E 2, for example, the regular polygon shown in Fig.1.2.1

in the next page, then it is clear that an isometry leaving X invariant must be a rotation

or a reflection, can not be a translation. In this case, the rotations that leave X invariant

are about the center of X through angles 2πi/n for n = 0, 1, 2, · · · , n − 1. The reflections

which preserve X are lines joining opposite vertices if n ≡ 0(mod2) (see Fig.1.2.1) or

lines through a vertex and the midpoint of the opposite edge if n ≡ 1(mod2).

Let ρ be a rotation about the center of X through angles 2π/n from the vertex 1

in counterclockwise and τ a reflection joining the vertex 1 with its opposite vertex if

n ≡ 0(mod2) or midpoint of its opposite edge if n ≡ 1(mod2).



10 Chap.1 Groups

...............................................................................

....................................................................................

................................................................................

......

n − 1 n

1n − 2

2πn

2

Fig.1.2.1

Then we know that ρn = 1 X , τ2 = 1 X , τ−1 ρτ = ρ−1.

We thereafter get the isometry group Dn of regular n-polygon to be

Dn = ρiτ j|0 ≤ i ≤ n − 1, 0 ≤ j ≤ 1.

This group is usually called the dihedral group of order 2n.

Definition 1.2.5 Let (G ; ) , (H ; ·) be groups. A bijection φ : G → H is an isomorphism

if

φ(a b) = φ(a) · φ(b)

for ∀a, b ∈ G . If such an isomorphism φ exists, the group (G ; ) is called to be isomorphic

to (H ; ·) , denoted by (G ; ) ≃ (H ; ·).

Example 1.2.5 Each group pair in the following is isomorphic.

(1) ( x ; ·), xn = 1 with (Zn; +);

(2) Klein 4-group in Table 2.2 with Z2 × Z2;

(3) GL(V ,R), dimV = n with (GL(n,R); ·).

1.2.3 Group Property. Elementary properties of groups are listed following.

P1. There is only one unit 1G in a group (G ; ).

In fact, if there are two units 1G and 1′G in (G ; ), then we get 1G = 1G 1′

G = 1′G , a

contradiction.



Sec.1.2 Groups 11

P2. There is only one inverse a−1 for a ∈ G in a group (G ; ).

If a−11 , a−1

2 both are the inverses of a ∈ G , then we get that a−11 = a−1

1 a a−12 = a−1

2 ,

a contradiction.

P3. (a−1)−1 = a, a ∈ G .

This is by the definition of inverse, i.e., a a−1 = a−1 a = 1G .

P4. If a b = a c or b a = c a, where a, b, c ∈ G , then b = c.

If a b = a c, then a−1 (a b) = a−1 (a c). According to the associative law, we

get that b = 1G b = (a−1 a) b = a−1 (a c) = (a−1 a) c = 1G c = c. Similarly, if

b a = c a, we can also get b = c.

P5. There is a unique solution for equations a

x = b and y

a = b in a group (G

;

) for

a, b ∈ G .

In fact, x = a−1 b and y = b a−1 are such solutions.

Denote by an = a a · · · a n

. Then the following property is obvious.

P6. For any integers n, m and a, b ∈ G , an am = an+m , (an)m = anm. Particularly, if (G ; )

is Abelian, then (a b)n = an bn.

Definition 1.2.6 Let (G ; ) be a group, a ∈ G . If there exists a least integer k ≥ 0 with

ak

= 1G , such k is called the order of a and denoted by o(a) = k. If there are no positive power of a equal to 1G , a has order infinity.

Theorem 1.2.1 Let (G ; ) be a group, x ∈ G and o( x) = k. Then

(1) xl = 1G if and only if k |l;

(2) if o( x) < +∞ , xl = xm if and only if k |l − m, and if o( x) = +∞ , then xl = xm if and

only if l = m.

Proof If k |l, let l = kd for an integer d . Then

xl = xkd = ( xk )d = 1d G = 1G .

Conversely, if k is not a divisor of l, let l = kd + r for integers d and r , 0 < r < k − 1.

Then we know that

xl = xkd +r = xkd xr = 1G xr 1G

by the definition of order. So we get (1).



12 Chap.1 Groups

Notice that xl = xm if and only if xl−m = 1G , i.e., l − m|k by (1). Furthermore, if

o( x) = +∞, then xl = xm only if l = m by definition. We get conclusion (2).

1.2.4 Subgroup. Let H be a subset of a group (G ;

). If (H ;

) is a group itself, then

it is called a subgroup of (G ; ), denoted by H ≤ G . If H ≤ G but H G , then H

is called a proper subgroup of G , denoted by H < G . We know a criterion of subgroups

following.

Theorem 1.2.2 Let H be a subset of a group (G ; ). Then (H ; ) is a subgroup of (G ; )

if and only if H ∅ and a b−1 ∈ H for ∀a, b ∈ H .

Proof By definition if (H ; ) is a group itself, then H ∅, there is b−1 ∈ H and

a b−1 is closed in H , i.e., a b−1 ∈ H for ∀a, b ∈ H .

Now if H ∅ and a b−1 ∈ H for ∀a, b ∈ H , then,

(1) there exists an h ∈ H and 1G = h h−1 ∈ H ;

(2) if x, y ∈ H , then y−1 = 1G y−1 ∈ H and hence x ( y−1)−1 = x y ∈ H ;

(3) the associative law x ( y z) = ( x y) z for x, y, z ∈ H is hold in (G ; ). By

(2), it is also hold in H . Thus (H ; ) is a group.

Corollary 1.2.1 Let H 1 ≤ G and H 2 ≤ G . Then H 1 ∩ H 2 ≤ G .

Proof Obviously, 1G = 1H 1 = 1H 2 ∈ H 1 ∩H 2. So H 1 ∩H 2 ∅. Let x, y ∈ H 1 ∩H 2.

Applying Theorem 1.2.2, we get that

x y−1 ∈ H 1, x y−1 ∈ H 2.

Whence,

x y−1 ∈ H 1 ∩ H 2.

Thus, (H 1 ∩ H 2; ) is a subgroup of (G ; ).

Let X be a subset of a group (G ; ). Define the subgroup X generated by X to be

the intersection of all subgroups of (G ; ) which contains X . Notice that there will be one

such subgroup, i.e., (G

;

) at least. So X

is a subgroup of (

G ;

) by Corollary 1.2.1. A

subgroup generated by one element x ∈ G ; ) is usually called a cyclic group, denoted by

x. The next result determines the form of each element in the subgroup X .

Theorem 1.2.3 Let X be a nonempty subset of a group (G ; ). Then X is the set of all

elements of the form xǫ 11

xǫ 22

· · · xǫ ss , where xi ∈ X, ǫ i = ±1 and s ≥ 0 (if s = 0 , this product

is interpreted to be 1G ).



Sec.1.2 Groups 13

Proof Let S denote the set of all such elements. Applying Theorem 1.2.2, we know

that (S ; ) is a subgroup of (G ; ). It is clear that X ⊂ S . Whence, X ⊂ S . But by

definition, it is obvious that S ⊂ X . So we get that S = X . .

For a finite subgroup H of (G ; ), the criterion of Theorem 1.2.2 can be simplified

to the following.

Theorem 1.2.4 Let H be a finite subset of a group (G ; ). Then (H ; ) is a subgroup of

(G ; ) if and only if H ∅ and a b ∈ H for ∀a, b ∈ H .

Proof The necessity is clear. We prove the sufficiency. By Theorem 1.2.2, we only

need to check b−1 ∈ H in this case. In fact, let b ∈ H . Then we get bm ∈ H for any

integer m ∈ Z + by assumption. But H is finite. Whence, there are integers k , l, k l

such that b

k

= b

l

. Not loss of generality, we assume k > l. Then b

k

−l

−1

= b−1

∈ H .Whence, (H ; ) is a subgroup of (G ; ).

Definition 1.2.7 Let (G , ) be a group, H ≤ G and a ∈ G . Define

a H = a h|h ∈ H and

H a = h a|h ∈ H ,

called the left or right coset of H , respectively.

Because the behavior of left coset is the same of that the right. We only discuss the

left coset following.

Theorem 1.2.5 Let H ≤ G with an operation and a, b ∈ G . Then

(1) for ∀b ∈ a H , a H = b H ;

(2) a H = b H if and only if b−1 a ∈ H ;

(3) a H = b H or a H ∩ b H = ∅.

Proof (1) If b ∈ a H , then there exists an element h ∈ H such that b = a h.

Therefore, b H = (a h) H = a (h H = a H .

(2) If a =b H , then there exist elements h1, h2 ∈ H such that a h1 = b h2.

Whence, b−1 a = h2 h−11 ∈ H . Conversely, if b−1 a ∈ H , then there exists h ∈ H

such that b−1 a = h, i.e., a ∈ bH . Applying the conclusion (1), we get aH = bH .

(3) In fact, if a H ∩ b H ∅, let c ∈ (a H ∩ b H ). Then, c H = a H

and c H = b H by the conclusion (1). Therefore, a H = b H .



14 Chap.1 Groups

Let us denote by G /H all these left (or right) cosets and G : H the resulting sets

by selecting an element from each left coset of H , called the left coset representation.

By Theorem 1.2.5, we get that

G = t ∈G :H

t H

and ∀g ∈ G can be uniquely written in the form t h for t ∈ G : H , h ∈ H . Usually,

|G : H | is called the index of H in G . For such indexes, we have a theorem following.

Theorem 1.2.6 (Lagrange) Let H ≤ G . Then |G | = |H ||G : H |.Proof Let

G =

t ∈G :H

t H .

Notice that t 1 H ∩ t 2 H = ∅ if t 1 t 2 and |t H | = |H |. We get that

|G | =

t ∈G :H

t H = |H ||G : H |.

Generally, we know the following theorem for indexes of subgroups. In fact, Theo-

rem 1.2.6 is just its a special case of K = 1K , the trivial group.

Theorem 1.2.7 Let K ≤ H ≤ G with an operation . Then (G : H )(H : K ) is a left

coset representation of K in G . Thus

|G : K | = |G : H ||H : K |.

Proof Let G =

t ∈G :H t H and H =

u∈H :K

u K . Whence,

G =

t ∈G :H , u∈H :K

t u K .

We show that all these cosets t u K are distinct. In fact, if t u K = t ′ u′ K for

some t , t ′

∈G : H , u, u′

∈H : K , then t −1

t ′

∈H and t

H = t ′

H by Theorem

1.2.5. By the uniqueness of left coset representations in G : H , we find that t = t ′.

Consequently, u K = u′ K . Applying the uniqueness of left coset representations in

H : K , we get that u = u′.

Let H ≤ G and K ≤ G with an operation . Define

H G = h g|h ∈ H , g ∈ G .



Sec.1.2 Groups 15

The subgroups H and K are said to be permute if H G = G H . Particularly, if for

∀g ∈ G , g H = H g, such subgroups H are very important, called the normal

subgroups of (G ; ), denoted by H ⊳ G .

Theorem 1.2.8 Let (G ; ) be a group and H ≤ G . Then the following three statements

are equivalent.

(1) x H = H x for ∀ x ∈ G ;

(2) x−1 H x = H for ∀ x ∈ G ;

(3) x−1 h x ∈ H for ∀ x ∈ G and h ∈ H .

Proof For (1) ⇒ (2), multiply both sides of (1) by x−1, we get (2). The (2) ⇒ (3)

is clear by definition. Now for (3) ⇒ (1), let h ∈ H and x ∈ G . Then we find that

h x = x ( x−1

h x) ∈ x H and x h = ( x−1

)−1

h x ∈ H x. Therefore, x H = H x.

Obviously, 1G ⊳ G and G ⊳ G . A group (G ; ) is called simple if there are no normal

subgroups diff erent from (1G ; ) and (G ; ) in (G ; ).

Although it is an arduous work for determining all subgroups, or normal subgroups

of a given group. But there is little difficulty in the case of cyclic groups.

Theorem 1.2.9 Let G = x and H ≤ G with an operation . Then

(1) if G is infinite, H is either infinite cyclic or trivial;(2) if G is finite, H is cyclic of order dividing n. Conversely, to each positive divisor

d of n, there is exactly one subgroup of order d, i.e., xn/d

.

Proof (1) If H is trivial, the conclusion is obvious. So let H 1H . Then there

is a minimal positive number k such that H contains some positive power xk 1H .

Obviously, xk

⊂ H . If xt ∈ H , we write t = kq + r , where 0 ≤ r ≤ k − 1. Then we find

that xr = ( xk )−q xt ∈ H . Contradicts the minimality of k . Whence, r = 0 and k |t . Hence

xt ∈

xk

and H =

xk

. If G is infinite, then x has infinite order, as does xk . Therefore,

H is also infinite.(2) Let o( x) = n. Then |H | divides n by Theorem 1.2.6. Conversely, suppose d |n.

Then o( xn/d ) = d and | xn/d

| = d . If there is another subgroup xs of order d . Then

xsd = 1H and n|sd . Consequently, we get n/d divides s. Whence, xs ≤ xn/d

. But they

both have the same order d , so xs = xn/d

.

Certainly, every subgroup of a cyclic group is normal. The following result com-



16 Chap.1 Groups

pletely determines simply cyclic groups.

Theorem 1.2.10 A cyclic group x is simple if and only if o( x) is prime.

Proof The sufficiency is immediately by Theorems 1.2.6 and 1.2.9. Moreover, xshould be finite. Otherwise, the subgroup

x2

would be its a normal subgroup, contradicts

to the assumption. By Theorem 1.2.9, we know that o( x) must be a prime number.

1.2.5 Symmetric Group. Let Ω = a1, a2, · · · , an be an n-set. A permutation on Ω is a

bijection σ : Ω → Ω. The cardinality |Ω| of Ω is called the degree of such a permutation

σ. Denoted by aσi the image of σ(ai) for 1 ≤ i ≤ n. Then σ can be also represented by

σ = a1 a2 · · · an

a

σ

1 a

σ

2 · · · a

σ

n .

Usually, we adopt Ω = 1, 2, · · · , n for simplicity. In this case, we represent σ by

σ =

1 2 · · · n

1σ 2σ · · · nσ

.

Let σ, τ be two permutations on Ω. The product στ is defined by

iστ = (σ)τ, for i = 1, 2, · · · , n.

For example, let

σ =

1 2 3 4

2 4 1 3

, τ =

1 2 3 4

2 1 4 3

.

Then we get that

στ =

1 2 3 4

2 4 1 3

1 2 3 4

2 1 4 3

=

1 2 3 4

1 3 2 4

.

Let σ be a permutation on Ω such that

aσ1 = a2, aσ

2 = a3, · · · , aσm−1 = am, aσ

m = a1

and fixes each element Ω \ a1, a2, · · · , am. We call such a permutation σ a m-cycle,

denoted it by (a1, a2, · · · , am) and its elements by [σ]. If m = 1, σ is the identity; if m = 2,

i.e., (a1, a2), such a σ is called involution.



Sec.1.2 Groups 17

Theorem 1.2.11 Any permutation σ can be written as a product of disjoint cycles, and

these cycles are unique.

Proof Let σ be a permutation on Ω =

1, 2,

· · ·, n

. Choose an element a

∈Ω.

Construct a sequence

a = aσ0

, aσ, aσ2

, · · · , aσk

, · · · ,

where aσk ∈ Ω for any integer k ≥ 0. Whence, there must be a least positive integer m

such that aσm

= aσi

, 0 ≤ i < m. Now if i 0, we get that (aσm−1

)σ = (aσi−1

)σ. But

aσm−1

aσi−1

by assumption. Whence, aσm

= (aσm−1

)σ (aσi−1

)σaσi

, a contradiction. So

i = 0, i.e., aσm

= a, or in other words, τ1 = (a, aσ, aσ2

, · · · , aσm−1

) is an m-cycle.

If Ω\[τ1] = ∅, then m = n and σ is an n-cycle. Otherwise, we can choose b ∈ Ω\[τ1]

and get a s-cycle τ2 = (b, bσ

, · · · , bσs

−1

).Similarly, if choose Ω \ ([τ1] ∪ [τ2] ∅, choose c in it and find a l-cycle τ3 =

(c, cσ, · · · , cσl−1

).

Continue this process. Because of the finiteness of Ω, we finally get an integer t and

cycles τ1, τ2, · · · , τt such that Ω \ ([τ1] ∪ [τ2] ∪ · · · ∪ [τt ] = ∅ and σ = τ1τ2 · · · τt with

disjoint cycles τi, 1 ≤ i ≤ t . The uniqueness of τi, 1 ≤ i ≤ t is clear by their construction.

Notice that

(a1, a2, · · · , am) = (a1, a2)(a1, a2) · · · (a1, am).

We can always represent a permutation by product of involutions by Theorem 1.2.11. For

example,

σ =

1 2 3 4 5

2 3 1 5 4

= (1, 2, 3)(4, 5)

= (1, 2)(1, 3)(4, 5) = (2, 3)(1, 2)(4, 5)

= (2, 3)(1, 2)(1, 3)(4, 5)(1, 3).

Definition 1.2.8 A permutation is odd (even) if it can be presented by a product of odd

(even) involutions.

Theorem 1.2.12 The property of odd or even of a permutation σ is uniquely determined

by σ itself.



18 Chap.1 Groups

Proof Let P be a homogeneous polynomial with form

P =

1≤i< j≤n

( xi − x j).

Clearly, any permutation leaves P unchanged as to its sign. For example, the involution

( x1 x2) changes ( x1− x2) into its negative ( x2− x1), interchanges ( x1− x j) with ( x2− x j), j > 2

and leaves the other factor unchanged. Whence, it changes P to −P. This fact means that

an odd (even) permutation σ always changes P to −P (P), only dependent on σ itself.

The next result is clear by definition.

Theorem 1.2.13 All permutations and all even permutations on Ω form groups, called

the symmetric group S Ω or alternating group AΩ , respectively.

Let τ, σ be permutations on Ω and σ = (a1, a2, · · · , am). A calculation shows that

τστ−1 = (aτ1, aτ

2, · · · , aτm).

Generally, if

σ = σ1σ2 · · · σs

is written a product of disjoint cycles for an integer s ≥ 1, Then

τστ−1 = σ′1σ′

2 · · · σ′s,

where the σ′i is obtained from σi replacing each entry a in σi by τ(a).

1.2.6 Regular Representation. Let (G ; ) be a group with

G = a1 = 1G , a2, · · · , an.

For ∀ai ∈ G , we know these elements

a1 ai, a2 ai, · · · , an ai

or

a−1i a1, a−1

i a2, · · · , a−1i an

still in G . Whence, they are both rearrangements of a1, a2, · · · , an. We get permutations

σai=

a1 a2 · · · an

a1 ai a2 ai · · · an ai

=

a

a ai

,



Sec.1.2 Groups 19

τai=

a1 a2 · · · an

a−1i

a1 a−1i

a2 · · · a−1i

an

=

a

a−1i

a

.

In this way, we get two sets of n permutations

RG = σa1, σa2

, · · · , σan and LG = τa1

, τa2, · · · , τan

.

Notice that each permutation ς in RG or LG is fixed-free, i.e., aς = a, a ∈ Ω only if ς = 1G .

We say RG , LG the right or left regular representation of G , respectively. The cardinality

|G | = n is called the degree of RG or LG .

Example 1.2.6 Let K = 1, α , β , γ be the Klein 4-group with an operation determined

by Table 1.2.2. Then we get elements σ1, σα, σ β, σγ in RK as follows.

σ1 = (1)(α)( β)(γ ),

σα =

1 α β γ

α 1 γ β

= (1, α)( β,γ ),

σ β =

1 α β γ

β γ 1 α

= (1, β)(α, γ ),

σγ = 1 α β γ

γ β α 1 = (1, γ )(α, β),

That is,

RK = (1)(α)( β)(γ ), (1, α)( β,γ ), (1, β)(α, γ ), (1, γ )(α, β).

Theorem 1.2.14 RG and LG both are subgroups of the symmetric group S G .

Proof Applying Theorem 1.2.4, we only need to prove that for two integers i, j, 1 ≤i, j ≤ n, σai

σa j∈ RG and τai

τa j∈ LG . In fact,

σaiσa j

= a

a ai a

a a j = a

a ai a j = σai

a j

∈RG ,

τaiτa j

=

a

a−1i a

a

a−1 j a

=

a

a−1 j a−1

i a

=

a

(ai a j)−1 a

= τaia j∈ LG .



20 Chap.1 Groups

Therefore, RG and LG both are subgroups of S G .

The importance of RG and LG are shown in the proof of next result.

Theorem 1.2.15(Cayley) Any group G is isomorphic to a subgroup of S G .

Proof Let (G ; ) be a group with G = a1 = 1G , a2, · · · , an. Define mappings

f : G → RG and h : G → LG by f (ai) = σai, h(ai) = τai

. Then f and h both are

one-to-one because of f (ai) f (a j), h(ai) h(a j) if ai a j. By the proof of Theorem

1.2.14, we know that

f (ai a j) = σaia j= σai

σa j= f (ai) f (a j),

h(ai a j) = τaia j= τai

τa j= h(ai)h(a j)

for integers 1 ≤ i, j ≤ n. So f and h are isomorphisms by definition. Consequently, G is

respective isomorphic to permutations RG and LG . Both of them are subgroups of S G by

Theorem 1.2.14.

§1.3 HOMOMORPHISM THEOREMS

1.3.1 Homomorphism. Let (G ; ), (G ′; ·) be groups. A mapping φ : G → G ′ is a

homomorphism if

φ(a b) = φ(a) · φ(b)

for ∀a, b ∈ G . A homomorphism φ is called to be a monomorphism or epimorphism if

it is one-to-one or surjective. Particularly, if φ is a bijection, such a homomorphism φ is

nothing but an isomorphism by definition.

Now let φ be a homomorphism. Define the image Imφ and kernel Kerφ respectively

as follows:

Imφ

≡G φ =

φ(g)

|g

∈G

,

Kerφ = g | φ(g) = 1G , g ∈ G .

For example, let (Z; +) and (Zn; +) be groups defined in Example 1.2.1. Define

φ : Z→ Zn by φ( x) = x(modn). Then φ is a surjection from (Z; +) to (Zn; +).

Let φ : G → H be a homomorphism. Some elementary properties of homomor-

phism are listed following.



Sec.1.3 Homomorphism Theorems 21

H1. φ( xn) = φn( x) for all integers n, x ∈ G , whence, φ(1G ) = 1H and φ( x−1) = φ−1( x).

By induction, this fact is easily proved for n > 0. If n = 0, by φ( x) = φ( x 1G ) =

φ( x)

·φ(1G ), we know that φ(1G ) = 1H . Now let n < 0. Then 1H = φ(1G ) = φ( xn

x−n) =

φ( xn) · φ( x−n), i.e., φ( xn) = φ−1( x−n) = (φ−n( x))−1 = φn( x).

H2. o(φ( x))|o( x) , x ∈ G .

In fact, Let o( x) = k . Then xk = 1G . Applying the property H1, we get that

φk ( x) = φ( xk ) = φ(1G ) = 1H .

By Theorem 1.2.1, we get that o(φ( x))|o( x).

The following property is obvious by definition.

H3. If x y = y x, then φ( x) · φ( y) = φ( y) · φ( x).

H4. Imφ ≤ H and Kerφ⊳ G .

This is an immediately conclusion of Theorems 1.2.2 and 1.2.8.

Theorem 1.3.1 A homomorphism φ : G → H is an isomorphism if and only if Kerφ =

1G .

Proof The necessity is clear. We prove the sufficiency. Let Kerφ = 1G . We prove

that φ is a bijection. If not, let φ( x) = φ( y) for two diff erent element x, y ∈ G , then

φ( x y−1) = φ( x) · φ−1( y) = 1H

by definition. Therefore, x y−1 ∈ Kerφ, i.e., x y−1 = 1G . Whence, we get x = y, a

contradiction.

1.3.2 Quotient Group. Let (G ; ) be a group, H 1, H 2, H 3 ≤ G . Define the multiplica-

tion and inverse of set by

H 1H 2 = x y | x ∈ H 1, y ∈ H 2 and H −11 = x−1 | x ∈ H 1 .

It is clear that H 1(H 2H 3) = (H 1H 2)H 3. By this definition, the criterion for a subset

H ⊂ G to be a subgroup of G can be written by

H H −1 ⊂ H .



22 Chap.1 Groups

Now we can consider this operation in G /H and determine when it is a group.

Generally, for ∀a, b ∈ G , we do not always get

(a

H )(b

H )

∈G /H

unless H ⊳ G . In fact, we have the following result for G /H .

Theorem 1.3.2 G /H is a group if and only if H is normal.

Proof If H is a normal subgroup, then

(a H )(b H ) = a (H b) H = a (b H ) H = (a b) H

by the definition of normal subgroup. This equality enables us to check laws of a group

following.

(1) Associative laws in G /H .

[(a H )(b H )](c H ) = [(a b) c] H = [a (b c)] H

= (a H )[(b H )(c H )].

(2) Existence of identity element 1G /H in G /H .

In fact, 1G /H = 1 H = H .

(3) Inverse element for

∀ x

H

∈G /H .

Because of ( x−1 H )( x H ) = ( x−1 x) H = H = 1G /H , we know the inverse

element of x H ∈ G /H is x−1 H .

Conversely, if G /H is a group, then for a H , b H ∈ G /H , we have

(a H )(b H ) = c H .

Obviously, a b ∈ (a H )(b H ). Therefore,

(a H )(b H ) = (a b) H .

Multiply both sides by a−1, we get that

H b H = b H .

Notice that 1G ∈ H , we know that

b H ⊂ H b H = b H ,



Sec.1.3 Homomorphism Theorems 23

i.e., b H b−1 ⊂ H . Consequently, we also find b−1 H b ⊂ H if replace b by b−1,

i.e., H ⊂ b H b−1. Whence,

b−1

H

b = H

for ∀b ∈ G . Namely, H is a normal subgroup of G .

Definition 1.3.1 If G /H is a group under the set multiplication, we say it is a quotient

group of G by H .

1.3.3 Isomorphism Theorem. If H is a normal subgroup of G , by Theorem 1.3.2 we

know that G /H is a group. In this case, the mapping φ : G → G /H determined by

φ( x) = x H is a homomorphism because

φ( x y) = ( x y) H = ( x H )( y H ) = φ( x)φ( y)

for all x, y ∈ G . It is clear that Imφ = G /H and Kerφ = H . Such a φ is called to be

natural homomorphism of groups. Generally, we know the following result.

Theorem 1.3.3(First Isomorphism Theorem) If φ : G → H is a homomorphism of

groups, then the mapping ς : x Kerφ → φ( x) is an isomorphism from G /Kerφ to Imφ.

Proof We have known that Kerφ ⊳ G by the property (H4) of homomorphism. So

G /Kerφ is a group by Theorem 1.3.2. Applying Theorem 1.3.1, we only need to check

that Kerς = 1G /Kerφ. In fact, x Kerφ ∈ Kerς if and only if x ∈ Kerφ. Thus ς is anisomorphism from from G /Kerφ to Imφ.

Particularly, if Imφ = H , we get a conclusion following, usually called the funda-

mental homomorphism theorem.

Corollary 1.3.1(Fundamental Homomorphism Theorem) If φ : G → H is an epimor-

phism, then G /Kerφ is isomorphic to H .

Theorem 1.3.4(Second Isomorphism Theorem) Let H ≤ G and N ⊳G . Then H ∩ N ⊳

G and x

(H

∩N )

→x

N is an isomorphism from H /H

∩N to H N /N .

Proof Clearly, the mapping τ : x → xN is an epimorphism from H to N H /N

with Kerτ = H ∩ N . Applying Theorem 1.3.3, we know that it is an isomorphism from

H /H ∩ N to H N /N .

Theorem 1.3.5(Third Isomorphism Theorem) Let M , N ⊳ G with N ≤ M . Then

M /N ⊳ G /N and (G /N )/(M /N ) ≃ G /M .



24 Chap.1 Groups

Proof Define a mapping ϕ : G /N → G /M by ϕ( x N ) = x M . Then

ϕ[( x N ) ( y N )] = ϕ[( x y) N ] = ( x y) M

= ( x M ) ( y M ) = ϕ( x N ) ϕ( y N )

and Kerϕ = M /N , Imϕ = G /M . So ϕ is an epimorphism. Applying Theorem 1.3.3,

we know that ϕ is an isomorphism from (G /N )/(M /N ) to G /M .

§1.4 ABELIAN GROUPS

1.4.1 Direct Product. An Abelian group is such a group (G ;

) with the commutative

law a b = b a hold for a, b ∈ G . The structure of such a group can be completely

characterized by direct product of subgroups following.

Definition 1.4.1 Let (G ; ) be a group. If there are subgroups A, B ≤ G such that

(1) for ∀g ∈ G , there are uniquely a ∈ A and b ∈ B such that g = a b;

(2) a b = b a for a ∈ A and b ∈ B, then we say (G ; ) is a direct product of A and

B, denoted by G = A ⊗ B.

Theorem 1.4.1 If G = A ⊗ B, then

(1) A⊳ G and B⊳ G ;

(2) G = AB;

(3) A ∩ B = 1G .

Conversely, if there are subgroups A, B of G with conditions (1)-(3) hold, then G = A ⊗ B.

Proof If G = A ⊗ B, by definition we immediately get that G = AB. If there is

c ∈ A ∩ B with c 1G , we get

c = c 1G , c ∈ A, 1G ∈ B

and

c = 1G c, 1G ∈ A, c ∈ B,

contradicts the uniqueness of direct product. So A ∩ B = 1G .



Sec.1.4 Abelian Groups 25

Now we prove A ⊳ G . For ∀a ∈ A, g ∈ G , by definition there are uniquely g1 ∈ A,

g2 ∈ B such that g = g1 g2. Therefore,

g−1

a

g = (g1

g2)−1

a

(g1

g2) = g−1

2

g−1

1

a

g1

g2

= g−11 a g1 g−1

2 g2 = g−11 a g1 ∈ A.

So A⊳ G . Similarly, we get B⊳ G .

Conversely, if there are subgroups A, B of G with conditions (1)-(3) hold, we prove

G = A ⊗ B. For ∀g ∈ G , by G = AB there are a ∈ A and b ∈ B such that g = a b. If there

are a′ ∈ A, b′ ∈ B also with g = a′ b′, then

a′−1 a = b′ b−1 ∈ A ∩ B.

But A∩ B = 1G . Whence, a′−1 a = b′ b−1 = 1G , i.e., a = a′ and b = b′. So the equality

g = a b is unique.

Now we prove a b = b a for a ∈ A and b ∈ B. Notice that A ⊳ G and B⊳ G , we

know that

a b a−1 b−1 = a (b a−1 b−1) ∈ A

and

a b a−1 b−1 = (a b a−1) b−1 ∈ B.

But A

∩B =

1G

. So

a b a−1 b−1 = 1G , i.e., a b = b a.

By Definition 1.4.1, we know that G = A ⊗ B.

Generally, we define the semidirect product of two groups as follows:

Definition 1.4.2 Let G and H be two subgroups of a group (T ; ) , α : H → AutG a

homomorphism. Define the semidirect product G ⋊α H of G and H respect to α to be

G ⋊α H =

(g, h)

|g

∈G , h

∈H

with operation · determined by

(g1, h1) · (g2, h2) = (g1 gα(h1)−1

2, h1 h2).

Clearly, if α is the identity homomorphism, then the semidirect product G ×α H is

nothing but the direct product G ⊗ H .



26 Chap.1 Groups

Definition 1.4.3 Let (G ; ) be a group. If there are subgroups A1, A2, · · · , As ≤ G such

that

(1) for

∀g

∈G , there are uniquely ai

∈Ai , 1

≤i

≤s such that

g = a1 a2 · · · as;

(2) ai a j = a j ai for a ∈ Ai and b ∈ A j , where 1 ≤ i, j ≤ s, i j, then we say (G ; )

is a direct product of A1, A2, · · · , As , denoted by

G = A1 ⊗ A2 ⊗ · · · ⊗ As.

Applying Theorem 1.4.1, by induction we can easily get the following result.

Theorem 1.4.2 If A1, A2, · · · , As ≤ G , then G = A1 ⊗ A2 ⊗ · · · ⊗ As if and only if

(1) Ai ⊳ G , 1 ≤ i ≤ s;

(2) G = A1 A2 · · · As;

(3) ( A1 · · · Ai−1 Ai+1 · · · As) ∩ Ai = 1G , 1 ≤ i ≤ s.

1.4.2 Basis. Let G = a1, a2, · · · , as be an Abelian group with an operation . If

ak 11

ak 22

· · · ak ss = 1G

for integers k 1, k 2, · · · , k s implies that ak ii

= 1G , i = 1, 2, · · · , s, then such a1, a2, · · · , as

are called a basis of the Abelian group (G ; ), denoted by B(G ) = a1, a2, · · · , as. The

following properties on basis of a group are clear by definition.

B1. If G = A ⊗ B and B( A) = a1, a2, · · · , as , B( B) = b1, b2, · · · , bt , then B(G ) =

a1, a2, · · · , as, b1, b2, · · · , bt .

B2. If B(G ) = a1, a2, · · · , as and A = a1, a2, · · · , al , B = al+1, al+2, · · · , as , where

1 < l < s, then G = A ⊗ B.

An importance of basis is shown in the next result.

Theorem 1.4.3 Any finite Abelian group has a basis.

Proof Let G = a1, a2, · · · , ar be an Abelian group with an operation . If r = 1,

then G is a cyclic group with a basis B(G ) = a1.

Assume our conclusion is true for generators less than r . We prove it is also true for

r generators. Let

ak 11

ak 22

· · · ak r r = 1G (1 − 1)




for integers k 1, k 2, · · · , k r . Define m = mink 1, k 2, · · · , k r . Without loss of generality, we

assume m = k 1. If m = 1, we find that

a1 = a−k 2

2 a

−k 3

3 · · · a−k r

r .

Hence, G = a2, a3, · · · , ar and the conclusion is true by the induction assumption.

So we can assume our conclusion is true for the power of a1 less than m and find

integers t i, si for i = 2, · · · , r such that

k i = t im + si, 0 ≤ si < m.

Let

a∗1 = a1 a

t 2

2 · · · a

t r

r . (1 − 2)

Substitute (1 − 2) into (1 − 1), we know that

(a∗1)m as2

2 · · · asr

r = 1G .

If there is an integer i, 1 ≤ i ≤ r such that si 0, then by the induction assumption, G has

a basis. So we can assume that

s2 = s3 = · · · = sr = 0

and get

(a∗1)m = 1G .

Notice that

a1 = a∗1 a−t 2

2 · · · a−t r

r .

Whence, G =

a∗1, a2, · · · , ar

. Now we prove that

G =

a∗

1

⊗ a2, · · · , ar .

For this objective, we only need to check thata∗

1

∩ a2, · · · , ar = 1G .

In fact, let a ∈

a∗1

∩ a2, · · · , ar . Then we know that

a = (a∗1)l = (a1 at 2

2 · · · at r

r )l = al2

2 · · · alr

r .



28 Chap.1 Groups

Therefore,

al1 at 2l−l2

2 · · · at r i−lr

r = 1G (1 − 3)

By the Euclidean algorithm, we can always find an integer d such that

0 ≤ l − dm < m.

By equalities (1 − 1) and (1 − 3), we get that

al−dm1 at 2l−l2−dm

2 · · · at r i−lr −dm

r = 1G .

By the induction assumption, we must have l − dm = 0. So

a = (a∗1)l = (a∗

1)dm = 1G .

Whence, we get that

G =

a∗1

⊗ a2, · · · , ar .

By the induction assumption again, let a2, · · · , ar = b2 ⊗ · · · ⊗ br . We know that

G =

a∗1

⊗ b2 ⊗ · · · ⊗ br .

This completes the proof.

Corollary 1.4.1 Any finite Abelian group is a direct product of cyclic groups.

1.4.3 Finite Abelian Group Structure. Theorem 1.4.3 enable us to know that a finite

Abelian group is the direct product of its cyclic subgroups. In fact, the structure of a

finite Abelian group is completely determined by its order. That is the objective of this

subsection.

Definition 1.4.4 Let p be a prime number, (G ; ) a group, g ∈ G and H ≤ G . Then g

is called a p-element, or H a p-subgroup if o(g) = pk or |H | = pl for some integers

k , l ≥ 0.

Definition 1.4.5 Let (G , ) be a group with |G | = pαn, ( p, n) = 1. Then each subgroup

H ≤ G with |H | = pα is called a Sylow p-subgroup of (G ; ).

Theorem 1.4.4 Let (G ; ) be a finite Abelian group with |G | = pα1

1pα2

2· · · pαs

s , where

p1, p2, · · · , ps are prime numbers, di ff erent two by two. Then

G = a1 ⊗ a2 ⊗ · · · ⊗ as




with o(ai) = pαi for 1 ≤ i ≤ s.

Proof By Corollary 1.4.1, a finite Abelian group is a direct product of cyclic groups,

i.e.,

G = a1 ⊗ a2 ⊗ · · · ⊗ ar .

If there is an integer i, 1 ≤ i ≤ r such that o(ai) is not a prime power, let o(ai) =

p β1

i1 p

β2

i2· · · p

βil

ilwith pi j

∈ pi, 1 ≤ i ≤ s, βi j> 0 for 1 ≤ j ≤ l. We prove that ai can be

uniquely written as ai = b1 b2 · · · bl such that o(b j) = p βi j

i j, bi b j = b j bi, 1 ≤ i, j ≤ l.

Now let o(ai) = m1m2 with (m1, m2) = 1. By a result in elementary number theory,

there are integers u1, u2 such that u1m1 + u2m2 = 1. Whence, au1m1+u2m2

i = au1m1

i au2m2

i =

au2m2

i au1m1

i. Choose c1 = au2m2

iand c2 = au1m1

i. Then cm1

1 = 1G and cm2

2 = 1G . Whence,

o(c1)

|m1, o(c2)

|m2. Because c1

c2 = c2

c1 and (o(c1), o(c2)) = 1, we know that m1m2 =

o(ai) = o(c1 c2) = o(c1)o(c2). So there must be o(c1) = m1 and o(c2) = m2. Repeating the

previous process, we finally get elements b1, b2, · · · , bl ∈ G such that ai = b1 b2 · · · bl

with o(b j) = p βi j

i j, bi b j = b j bi, 1 ≤ i, j ≤ l.

Whence, we can assume that the order of each cyclic group in the direct product

G = a1 ⊗ a2 ⊗ · · · ⊗ ar .

is a prime power. Now if the order of

ai1

,

ai2

,· · ·, aik

are all with a same base pi,

replacing ai1 ai2

· · · aik by ai we get a direct product

G = a1 ⊗ a2 ⊗ · · · ⊗ as

with o(ai) = pαi

i, 1 ≤ i ≤ l.

Theorem 1.4.5 Let (G ; ) be a finite Abelian p-group. If

G = A1 ⊗ A2 ⊗ · · · ⊗ Ar and G = B1 ⊗ B2 ⊗ · · · ⊗ Bs,

where Ai, B j are cyclic p-groups for 1 ≤ i ≤ r, 1 ≤ j ≤ s, then s = r and there is a

bijection : A1, A2, · · · , Ar → B1, B2, · · · , Br such that | Ai| = | ( Ai)| , 1 ≤ i ≤ r.

Proof We prove this result by induction on |G |. If |G | = p, the conclusion is clear.

Define G p = a ∈ G |a p = 1G and G p = a p|a ∈ G . Notice that

G = A1 ⊗ A2 ⊗ · · · ⊗ Ar .

If ai ∈ Ai is the generator of Ai, 1 ≤ i ≤ r , then B(G ) = a1, a2, · · · , ar . Let o(ai) = pei .

Without loss of generality, we can assume that e1 ≥ e2 ≥ · · · ≥ er ≥ 1. Then B(G p) =



30 Chap.1 Groups

a pe1−1

1, a

pe2−1

2, · · · , a

per −1

r and |G p| = pr . If e1 = e2 = · · · = er = 1, then G p = 1G .

Otherwise, let e1 ≥ e2 ≥ · · · ≥ em > em+1 = · · · = er = 1. Then B(G p) = a p

1, a

p

2, · · · , a

pm.

Now let bi ∈ Bi be its a generator for 1 ≤ i ≤ s. Then B(G ) = b1, b2, · · · , bs. Let

o(bi) = p f i , 1 ≤ i ≤ s with f 1 ≥ f 2 ≥ · · · ≥ f s. Similarly, we know that |G p| = ps. So s = r .Now if G p = 1G , there must be f 1 = f 2 = · · · = f s = 1. Otherwise, if G p 1G , let

f 1 ≥ f 2 ≥ · · · ≥ f m′ > f m′+1 = · · · = f s = 1. Then B(G p) = b p

1, b

p

2, · · · , b

p

m′. Notice that

|G p| < |G |, by the induction assumption, we get that m = m′ and ei = f i for 1 ≤ i ≤ r .

Therefore, o(ai) = o(bi) for 1 ≤ i ≤ r . Now define : A1, A2, · · · , Ar → B1, B2, · · · , Br by ( Ai) = Bi, 1 ≤ i ≤ r . We get | Ai| = | ( Ai)| for integers 1 ≤ i ≤ r .

Combining Theorems 1.4.4 and 1.4.5, we get the fundamental theorem of finite

Abelian groups following.

Theorem 1.4.6 Any finite Abelian group (G ; ) is a direct product

G = a1 ⊗ a2 ⊗ · · · ⊗ as

of cyclic p-groups uniquely determined up to their cardinality.

These cardinalities | a1 |, | a2 |, · · · , | as | in Theorem 1.4.6 are defined to be the

invariants of Abelian group (G ; ), denoted by InvarG . Then we immediately get the

following conclusion by Theorem 1.4.6.

Corollary 1.4.2 Let G , H be finite Abelian groups. Then G ≃ H if and only if InvarG =

InvarH .

§1.5 MULTIGROUPS

1.5.1 MultiGroup. Let

G be a set with binary operations

O. A pair (

G ;

O) is an algebraic

multi-system if for

∀a, b

∈ G and

∈ O, a

b

∈ G provided a

b existing.

We consider algebraic multi-systems in this section.

Definition 1.5.1 For an integer n ≥ 1 , an algebraic multi-system ( G ; O) is an n-multigroup

if there are G 1, G 2, · · · , G n ⊂ G , O = i, 1 ≤ i ≤ n with

(1) G =n

i=1

G i;

(2) (G i; i) is a group for 1 ≤ i ≤ n.



Sec.1.5 Multi-Groups 31

For ∀ ∈ O, denoted by G the group (G ; ) and G max the maximal group (G ; ), i.e.,

(G max ; ) is a group but (G max

∪ x; ) is not for ∀ x ∈ G \ G max in ( G ; O).

Definition 1.5.2 Let ( G 1; O1) and ( G 2; O2) be multigroups. Then ( G 1; O1) is isomorphic to

( G 2; O2) , denoted by (ϑ, ι) : ( G 1; O1) → ( G 2; O2) if there are bijections ϑ : G 1 → G 2 and

ι : O1 → O2 such that for a, b ∈ G 1 and ∈ O1 ,

ϑ(a b) = ϑ(a)ι()ϑ(b)

provided ab existing in ( G 1; O1). Such isomorphic multigroups are denoted by ( G 1; O1) ≃( G 2; O2)

Clearly, if (

G 1;

O1) is an n-multigroup with (ϑ, ι) an isomorphism, the image of (ϑ, ι)

is also an n-multigroup. Now let (ϑ, ι) : ( G 1; O1) → ( G 2; O2) with G 1 =

ni=1 G 1i, G 2 =

ni=1 G 2i,O1 = 1i, 1 ≤ i ≤ n and O2 = 2i, 1 ≤ i ≤ n, then for ∈ O, G max is isomorphic to

ϑ(G )maxι() by definition. The following result shows that its converse is also true.

Theorem 1.5.1 Let ( G 1; O1) and ( G 2; O2) be n-multigroups with

G 1 =

ni=1

G 1i, G 2 =

ni=1

G 2i,

O1 =

i1, 1

≤i

≤n

, O2 =

i2, 1

≤i

≤n

. If φi : G 1i

→G 2i is an isomorphism for each

integer i, 1 ≤ i ≤ n with φk |G 1k ∩G 1l= φl|G 1k ∩G 1l

for integers 1 ≤ k , l ≤ n, then ( G 1; O1) is

isomorphic to ( G 2; O2).

Proof Define mappings ϑ : G 1 → G 2 and ι : O1 → O1 by

ϑ(a) = φi(a) if a ∈ G i ⊂ G and ι(1i) = 2i for each integer 1 ≤ i ≤ n.

Notice that φk |G 1k ∩G 1l= φl|G 1k ∩G 1l

for integers 1 ≤ k , l ≤ n. We know that ϑ, ι both are

bijections. Let a, b ∈ G 1s for an integer s, 1 ≤ s ≤ n. Then

ϑ(a 1s b) = φs(a 1s b) = φs(a) 2s φs(b) = ϑ(a)ι(1s)ϑ(b).

Whence, (ϑ, ι) : ( G 1; O1) → ( G 1; O1).

1.5.2 Submultigroup. Let ( G ; O) be a multigroup, H ⊂ G and O ⊂ O. If ( H ; O) is

multigroup itself, then (H ; O) is called a submultigroup, denoted by ( H ; O) ≤ ( G ; O).

Then the following criterion is obvious for submultigroups.



32 Chap.1 Groups

Theorem 1.5.2 An multi-subsystem ( H ; O) of a multigroup ( G ; O) is a submultigroup if

and only if H ∩ G ≤ G max for ∀ ∈ O.

Proof By definition, if ( H ; O) is a multigroup, then for

∀ ∈O, H

∩G is a group.

Whence, H ∩ G ≤ G max .

Conversely, if H ∩ G ≤ G max for ∀ ∈ O, then H ∩ G is a group. Therefore,

( H ; O) is a multigroup by definition.

Applying Theorem 1.2.2, we get corollaries following.

Corollary 1.5.1 An multi-subsystem ( H ; O) of a multigroup ( G ; O) is a submultigroup

if and only if a b−1 ∈ H ∩ G max for ∀ ∈ O and a, b ∈ H provided a b existing in

(

H ; O).

Particularly, if O = , we get a conclusion following.

Corollary 1.5.2 Let ∈ O. Then (H ; ) is submultigroup of a multigroup ( G ; O) for

H ⊂ G if and only if (H ; ) is a group, i.e., a b−1 ∈ H for a, b ∈ H .

A multigroup ( G ; O) is said to be a symmetric n-multigroup if there are S 1, S 2,

· · · , S n ⊂ G , O = i, 1 ≤ i ≤ n with

(1) G =n

i=1

S i;

(2) (S i; i) is a symmetric group S Ωifor 1 ≤ i ≤ n. We call the n-tuple (|Ω1|, |Ω2|, · · · , |Ωn||)

the degree of the symmetric n-multigroup ( G ; O).

Now let multigroup ( G ; O) be a n-multigroup with G 1, G 2, · · · , G n ⊂ G , O = i, 1 ≤i ≤ n. For any integer i, 1 ≤ i ≤ n. Let G i

= ai1 = 1G i, ai2, · · · , aini

. For ∀aik ∈ G i,

define

σaik =

ai1 ai2 · · · ain

ai1 aik ai2 aik · · · aini aik

=

a

a aik

,

τaik = ai1 ai2

· · ·ain

i

a−1ik

ai1 a−1ik

ai2 · · · a−1ik

aini

= a

a−1ik

a Denote by RG i = σai1

, σai2, · · · , σaini

and LG i = τai1, τai2

, · · · , τaini and ×r

i or ×li

the

induced multiplication in RG i or LG i . Then we get two sets of permutations

R G =

ni=1

σai1, σai2

, · · · , σaini and LG =

ni=1

τai1, τai2

, · · · , τaini.




We say R G , L G the right or left regular representation of G , respectively. Similar to

Theorem 1.2.15, the Cayley theorem, we get the following representation result for multi-

groups.

Theorem 1.5.3 Every multigroup is isomorphic to a submultigroup of symmetric multi-

group.

Proof Let multigroup ( G ; O) be a n-multigroup with G 1, G 2, · · · , G n ⊂ G , O =

i, 1 ≤ i ≤ n. For any integer i, 1 ≤ i ≤ n. By Theorem 1.2.14, we know that

RG i and LG i both are subgroups of the symmetric group S G i for any integer 1 ≤ i ≤ n.

Whence, ( R G ; Or ) and ( L G ; Ol) both are submultigroup of symmetric multigroup by defi-

nition, where Or = ×r i|1 ≤ i ≤ n and Ol = ×l

i|1 ≤ i ≤ n.

We only need to prove that ( G ; O) is isomorphic to ( R G ; Or ). For this objective,

define a mapping ( f , ι) : ( G ; O) → ( R G ; Or ) by

f (aik ) = σaik and ι(i) = ×r

i

for integers 1 ≤ i ≤ n. Such a mapping is one-to-one by definition. It is easily to see that

f (ai j i aik ) = σai jiaik = σai j

×r i σaik

= f (ai j)ι(i) f (aik )

for integers 1 ≤ i, k , l ≤ n. Whence, ( f , ι) is an isomorphism from (

G ;

O) to ( R

G ; Or ).

Similarly, we can also prove that ( G ; O) ≃ ( L G ; Ol

).

1.5.3 Normal Submultigroup. A submultigroup ( H ; O) of ( G ; O) is normal, denoted

by ( H ; O)⊳ ( G ; O) if for ∀g ∈ G and ∀ ∈ O

g H = H g,

where g H = g h|h ∈ H provided g h existing and H g is similarly defined.

Then we get a criterion for normal submultigroups of a multigroup following.

Theorem 1.5.4 Let ( H ; O)≤

( G ; O). Then ( H ; O)⊳ ( G ; O) if and only if

H ∩ G max ⊳ G max

for ∀ ∈ O.

Proof If H ∩ G max ⊳ G max

for ∀ ∈ O, then g H = H g for ∀g ∈ G max by

definition, i.e., all such g ∈ G and h ∈ H with gh and hg defined. So ( H ; O)⊳( G ; O).



34 Chap.1 Groups

Now if ( H ; O)⊳ ( G ; O), it is clear that H ∩ G max ⊳ G max

for ∀ ∈ O.

For a normal submultigroup (

H ; O) of (

G ;

O), we know that

(a H )(b · H ) = ∅ or a H = b · H .

In fact, if c ∈ (a H )

(b · H ), then there exists h1, h2 ∈ H such that

a h1 = c = b · h2.

So a−1 and b−1 exist in G max and G max

· , respectively. Thus,

b−1 · a h1 = b−1 · b · h2 = h2.

Whence,b−1 · a = h2 h−1

1 ∈ H .

We find that

a H = b · (h2 h1) H = b · H .

This fact enables one to find a partition of G following

G =

g∈

G ,∈

O

g

H .

Choose an element h from each g H and denoted by H all such elements, called the

representation of a partition of G , i.e.,

G =

h∈ H ,∈Oh H .

Define the quotient set of G by H to be

G / H = h H |h ∈ H , ∈ O.

Notice that H is normal. We find that

(a H ) · (b • H ) = H a · b • H = (a · b) H • H = (a · b) H

in G / H for , •, · ∈ O, i.e., ( G / H ; O) is an algebraic system. It is easily to check that

( G / H ; O) is a multigroup by definition, called the quotient multigroup of G by H .




Now let ( G 1; O1) a n d ( G 2; O2) be multigroups. A mapping pair (φ, ι) with φ : G 1 → G 2

and ι : O1 → O2 is a homomorphism if

φ(a

b) = φ(a)ι(

)φ(b)

for ∀a, b ∈ G and ∈ O1 provided a b existing in ( G 1; O1). Define the image Im(φ, ι)

and kernel Ker(φ, ι) respectively by

Im(φ, ι) = φ(g) | g ∈ G 1 ,

Ker(φ, ι) = g | φ(g) = 1G , g ∈ G 1 , ∈ O2.

Then we get the following isomorphism theorem for multigroups.

Theorem 1.5.5 Let (φ, ι) : ( G 1; O1) → ( G 2; O2) be a homomorphism. Then

G 1/Ker(φ, ι) ≃ Im(φ, ι).

Proof Notice that Ker(φ, ι) is a normal submultigroup of ( G 1; O1). We prove that the

induced mapping (σ, ω) determined by (σ, ω) : x Ker(φ, ι) → φ( x) is an isomorphism

from G 1/Ker(φ, ι) to Im(φ, ι).

Nowif(σ, ω)( x1) = (σ, ω)( x2), then we get that (σ, ω)( x1 x−12 ) = 1G provided x1 x−1

2

existing in ( G 1; O1), i.e., x1

x−1

∈Ker(φ, ι). Thus x1

Ker(φ, ι) = x2

Ker(φ, ι), i.e., the

mapping (σ, ω) is one-to-one. Whence it is a bijection from G 1/Ker(φ, ι) to Im(φ, ι).

For ∀a Ker(φ, ι), b Ker(φ, ι) ∈ G 1/Ker(φ, ι) and · ∈ O1, we get that

(σ, ω)[a Ker(φ, ι) · b • Ker(φ, ι)]

= (σ, ω)[(a · b) Ker(φ, ι)] = φ(a · b)

= φ(a)ι(·)φ(b) = (σ, ω)[a Ker(φ, ι)]ι(·)(σ, ω)[b • Ker(φ, ι)].

Whence, (σ, ω) is an isomorphism from

G 1/Ker(φ, ι) to Im(φ, ι).

Particularly, let ( G 2; O2) be a group in Theorem 1.5.4, we get a generalization of thefundamental homomorphism theorem, i.e., Corollary 1.3.1 following.

Corollary 1.5.3 Let ( G ; O) be a multigroup and (ω, ι) : ( G ; O) → (A ; ) an epimorphism

from ( G ; O) to a group (A ; ). Then

G /Ker(ω, ι) (A ; ).



36 Chap.1 Groups

1.5.4 Abelian Multigroup. For an integer n ≥ 1, an n-multigroup ( G ; O) is Abelian if

there are A 1, A 2, · · · , A n ⊂ G , O = i, 1 ≤ i ≤ n with

(1) G =n

i=1

A i;

(2) (A i; i) is Abelian for integers 1 ≤ i ≤ n.

For ∀ ∈ O, a commutative set of G max is defined by

C (G ) = a, b ∈ G max |a b = b a.

Such a set is called maximal if C (G ) ∪ x for x ∈ G max \ C (G ) is not commutative again.

Denoted by Z max(G ) the maximal commutative set of G max . Then it is clear that Z max(G )

is an Abelian subgroup of G max .

Theorem 1.5.6 An n-multigroup ( G ; O) is Abelian if and only if there are Z max(G ) for

∀ ∈ O such that G =∈O

Z max(G ).

Proof If G =∈O Z max(G ), it is clear that ( G ; O) is Abelian since Z max(G ) is an

Abelian subgroup of G max . Now if ( G ; O) is Abelian, then there are A 1, A 2, · · · , A n ⊂ G ,

O = i, 1 ≤ i ≤ n such that

G =

ni=1

A i

and ( Ai; i) is an Abelian group for 1 ≤ i ≤ n. Whence, there exists a maximal commuta-

tive set Z max(G i) ⊂ G max

such that Ai ⊂ Z max(G i). Consequently, we get that

G =

ni=1

Z max(G i).


Combining Theorems 1.5.6 with 1.4.6, we get the structure of finite Abelian multi-

group following.

Theorem 1.5.7 A finite multigroup ( G ; O) is Abelian if and only if there are generators

ai , 1 ≤ i ≤ s for ∀ ∈ O such that

G =∈O

a1

⊗ a2

⊗ · · · ⊗

as

.




1.5.5 Bigroup. A bigroup is nothing but a 2-multigroup. There are many examples of

bigroups in algebra. For example, these natural number field (Q; +, ·), real number num-

ber field (R; +, ·) and complex number field (C; +, ·) are all Abelian bigroups. Generally,

a field (F ; +, ·) is an algebraic system F with two operations +, · such that

(1) (F ; +) is an Abeilan group with identity 0;

(2) (F \ 0; ·) is an Abelian group;

(3) a · (b + c) = a · b + a · c for ∀a, b, c ∈ F .

Thus a field is an Abelian 2-group with an additional condition (3) called the dis-

tributive law following.

Definition 1.5.3 A bigroup (C ; +, ·) is distributive if

a · (b + c) = a · b + a · c

hold for all a, b, c ∈ B.

Theorem 1.5.8 Let (C ; +, ·) be a distributive bigroup of order ≥ 2 with C = A1 ∪ A2 such

that ( A1; +) and ( A2, ·) are groups. Then there must be A1 A2.

Proof Denoted by 0+, 1· the identities in groups ( A1; +), ( A2, ·), respectively. If

A1 = A2 = C , we get 1+, 1· ∈ A1 and A2. Because ( A2, ·) is a group, there exists an inverse

element 0−1

+ in A2 such that 0−1

+ · 0+ = 1·. By the distributive laws, we know that

a · 0+ = a · (0+ + 0+) = a · 0+ + a · 0+

hold for ∀a ∈ C . Whence, a · 0+ = 0+. Particularly, let a = 0−1+ , we get that 0−1

+ · 0+ = 0+,

which means that 0+ = 1·. But if so, we must get that

a = a 1 = a 0+ = 0+,

contradicts to the assumption |C | ≥ 2.

Theorem 1.5.8 implies the following conclusions.

Corollary 1.5.3 Let (G ; ) be a non-trivial group. Then there are no operations · on

G such that (G ; , ·) is a distributive bigroup.

Corollary 1.5.4 Any bigroup (C ; , ·) of order ≥ 2 with groups (C ; ) and (C , ·) is not

distributive.



38 Chap.1 Groups

Corollary 1.5.4 enables one to classify bigroups into the following categories:

Class 1. (1C ; +, ·) , i.e., which is a union of two trivial groups (1C ; +) and (1C ; ·).

Class 2. Non-distributive bigroups of order ≥ 2.

This kind of bigroup is easily found. Let (G 1; ) and (G 2; ·) be two groups without

the definition a b · c and a · b for a, b, c ∈ C , where C = G 1 ∪ G 2. Then (C ; , ·) is a

non-distributive bigroup with order≥ 2.

Class 3. Distributive bigroups of order ≥ 2.

In fact, any field is such a distributive Abelian bigroup. Certainly, we can find a more

general result for the existence of finite distributive bigroups.

Theorem 1.5.9 There are finite distributive Abelian bigroups (C ; +, ·) of order ≥ 2 with

groups ( A1; +) and ( A2, ·) such that C = A1 ∪ A2 for | A1 − A2| = |C | −m, where (m + 1)||C |.

Proof In fact, let (F ; +, ·) be a field. Then (F ; +) and (F \ 0+; ·) both are Abelian

group. Applying Theorem 1.4.6, we know that there are subgroups ( A′2; ·) of (F \ 0+; ·)

with order m, where (m+1)||C |. Obviously, C = A1 ∪ A′2. So (F ; +, ·) is also a distributive

Abelian bigroup with groups ( A1; +) and ( A′2

, ·) such that C = A1 ∪ A2 and | A1 − A2| =

|C

| −m.

A group (H ; ) (or (H ; ·)) is maximum in a bigroup (G ; , ·) if there are no groups

(T ; ) (or (T ; ·)) in (G ; , ·) such that |H | < |T |. Combining Theorem 1.5.9 with Corol-

laries 1.5.3 and 1.5.4, we get the following result on fields.

Theorem 1.5.10 A field (F ; +, ·) is a distributive Abelian bigroup with maximum groups

(F ; +) and (F \ 0+; ·).

1.5.6 Constructing Multigroup. There are many ways to get multigroups. For example,

let G be a set. Define n binary operations

1,

2,

· · ·,

n such that (G ;

i) is a group for any

integer i, 1 ≤ i ≤ n. Then (G ; i, 1 ≤ i ≤ n) is a multigroup by definition. In fact, the

structure of a multigroup is dependent on its combinatorial structure, i.e., its underlying

graph, which will be discussed in Chapter 3. In this subsection, we construct multigroups

only by one group or one field.

Construction 1.5.1 Let (G ; ) be a group and S G the symmetric group on G . For ∀a, b ∈



Sec.1.6 Remarks 39

G and

ω =

a

aω

∈ S G ,

define a binary operation ω on G ω

= G by

a ω b = (aω−1 bω−1

)ω

for ∀a, b ∈ G , Clearly, (G ω; ω) is a group and ω : (G ; ) → (G ω; ω) is an isomorphism.

Now for an integer n ≥ 1, choose n permutations ω1, ω2, · · · , ωn. Then we get a multi-

group (G ; ωi|1 ≤ i ≤ n), where groups (G ; ωi

) is isomorphic to (G ; ω j) for integers

1 ≤ i, j ≤ n. Therefore, we get the following result of multigroups.

Theorem 1.5.11 There is a multigroup P such that each of its group is isomorphic to

others in P.

Construction 2.5.2 Let (F ; +, ·) be a field and S F the symmetric group acting on F .

For ∀c, d ∈ G and ω ∈ S F , define a binary operation ω on F ω = F by

a +ω b = (aω−1

+ bω−1

)ω

and

a

·ω b = (aω−1

·bω−1

)ω

for ∀a, b ∈ G . Choose n permutations ς 1, ς 2, · · · , ς n ∈ S F . Then we get a multigroup

F = (F ; +ς i , 1 ≤ i ≤ n, ·ς i, 1 ≤ i ≤ n),

which enables us immediately to get a result following.

Theorem 1.5.12 There is a multigroup (F ; +i , 1 ≤ i ≤ n, ·i ; 1 ≤ i ≤ n) such that

for any integer i, (F ; +i, ·i) is a field and it is isomorphic to (F ; + j, · j) for any integer

j, 1

≤i, j

≤n.

§1.6 REMARKS

1.6.1 There are many standard books on abstract groups, such as those of [BiM1], [Rob1],

[Wan1], [Xum1] and [Zha1] for examples. In fact, the materials in Sections 1.1-1.4 are



40 Chap.1 Groups

mainly extracted from references [BiM1] and [Wan1] as an elementary introduction to

groups.

1.6.2 For an integer n

≥1, a Smarandache multi-space is a union of spaces A1, A2,

· · ·, An

diff erent two by two. Let Ai, 1 ≤ i ≤ n be mathematical structures appeared in sciences,

such as those of groups, rings, fields, metric spaces or physical fields, we therefore get

multigroups, multrings, multfields, multmetric spaces or physical multi-fields. The mate-

rial of Section 1.5 is on multigroups with new results. More results on multi-spaces can be

found in references [Mao4]-[Mao10], [Mao20], [Mao24]-[Mao25] and [Sma1]-[Sma2].

1.6.3 The conceptions of bigroup and sub-bigroup were first appeared in [Mag1] and

[MaK1]. Certainly, they are special cases of multigroup and submultigroup, i.e., special

cases of Smarandache multi-spaces. More results on bigroups can be found in [Kan1].In fact, Theorems 1.5.2-1.5.5 are the generalization of results on bigroups appeared in

[Kan1].

1.6.4 The applications of groups to other sciences are mainly by surveying symmetries of

objects, i.e., the action groups. For this objective, an elementary introduction has been ap-

peared in Subsection 1.2.6, i.e., regular representation of group. In fact, those approaches

can be only surveying global symmetries of objects. For locally surveying symmetries,

we are needed locally action groups, which will be introduced in the following chapter.



CHAPTER 2.

Action Groups

Action groups, i.e., group actions on objects are the oldest form, also theorigin of groups. The action idea enables one to measure similarity of ob-

jects, classify algebraic systems, geometrical objects by groups, which is the

fountain of applying groups to other sciences. Besides, it also allows one to

find symmetrical configurations, satisfying the aesthetic feeling of human be-

ings. Topics covered in this chapter including permutation groups, transitive

groups, multiply transitive groups, primitive and non-primitive groups, auto-

morphism groups of groups and p-groups. Generally, we globally measure

the symmetry of an object by group action. If allowed the action locally, then

we need the conception of locally action group, i.e., action multi-group, a

generalization of group actions to multi-groups discussed in this chapter.



42 Chap.2 Action Groups

§2.1 PERMUTATION GROUPS

2.1.1 Group Action. Let (G ; ) be a group and Ω = a1, a2, · · · , an. By a right action of

G on Ω is meant a mapping ρ : Ω

×G

→Ω such that

( x, g1 g2) ρ = (( x, g1) ρ, g2) ρ and ( x, 1G ) ρ = x.

It is more convenient to write xg instead of ( x, g) ρ. Then the defining equations become

xg1 g2 = ( xg1 )g2 and x1G = x, x ∈ Ω, g1, g2 ∈ G .

For a fixed g ∈ G , the inverse mapping of x → xg is x → xg−1

. Whence. x → xg is a

permutation of Ω. Denote this permutation by gγ . Then (g1 g2)γ maps x to xg1g2 , as does

gγ

1g

γ

2. We find that (g

1 g

2)

γ = gγ

1g

γ

2. Therefore, the group action determines a homomor-phism γ : G → S Ω. Such a homomorphism γ is called a permutation representation of G

on Ω.

Two permutation representations of a group γ : G → S X and δ : G → S X of a group

G on X and Y are said to be equivalent if there exists a bijection θ : X → Y such that

θ gδ = gγ θ, i.e., xθ gδ

= xgγ θ

for all x ∈ X and g ∈ G . Particularly, if X = Y , then there are some θ ∈ S X such that

gδ = θ −

1gγ θ . Certainly, we do not distinguish equivalent representations of permutation

groups in the view of action.

Let γ : G → S Ω be a permutation representation of G on Ω. The cardinality of

Ω is called the degree of this representation. A permutation representation is faithful if

Kerγ = 1G . So the subgroups P of S Ω are particularly important, called permutation

groups. For a ∈ Ω and τ ∈ P, we usually denote the image of a under τ by aτ,

τ =

a1 a2 · · · an

aτ1

aτ2

· · ·aτ

n

=

a

aτ

.

As a special case of equivalent representations of groups, let P1 and P2 be two

permutation groups action on Ω1, Ω2, respectively. A similarity from P1 to P2 is a pair

(γ, θ ) consisting of an isomorphism γ : P1 → P2 and a bijection θ : Ω1 → Ω2 which are

related by

πθ = θπγ , i.e., aπθ = aθπγ



Sec.2.1 Permutation Groups 43

for all a ∈ Ω1 and π ∈ P1. Particularly, if Ω1 = Ω2, this equality means that πγ = θ −1πθ

for ∀π, θ ∈ for ∀π ∈ P1.

2.1.2 Stabilizer. The stabilizer Pa and orbit aP of an element a in P are respectively

defined as follows:

Pa = σ | aσ = a, σ ∈ P and aP = b | aσ = b, σ ∈ P .


Theorem 2.1.1 Let P be a permutation group acting on Ω , x, y ∈ P and a, b ∈ Ω. Then

(1) aP ∩ bP = ∅ or aP = bP , i.e., all orbits forms a partition of Ω;

(2)Pa

is a subgroup of P

and if b = a x , thenPb

= x−1

Pa x. Moreover, if a x = b y ,

then xPa = yPa;

(3) |aP | = |P : Pa| , particularly, if P is finite, then |P | = |Pa||aP | for ∀a ∈ Ω.

Proof If c ∈ aP , then there is z ∈ P such that c = a z. Whence,

cP = c x| x ∈ P = a zx | x ∈ P = aP .

So aP ∩ bP = ∅ or aP = bP . Notice that an element a ∈ P lies in at least one obit aP ,

we know that all obits forms a partition of the set Ω. This proves (1).

For (2), it is clear that 1P ∈ Pa and for x, y ∈ Pa, xy−1 ∈ Pa. So Pa is a subgroup

of P by Theorem 1.2.2. Now if b = a x, then we know that

y ∈ Pb ⇔ a xy = a x ⇔ xyx−1 ∈ Pa,

i.e., y ∈ x−1Pa x, Whence, x−1P x = Pb. Finally,

a x = a y ⇔ a xy−1

= a ⇔ xy−1 ∈ Pa ⇔ xPa = yPa.

So (2) is proved.

Applying the conclusion (2), we know that there is a bijection between the distinctelements in aP and right cosets of Pa in P. Therefore |aP | = |P : Pa|. Particularly, if

P is finite, then |aP | = |P : Pa| = |P |/|Pa|. So we get that |P | = |Pa||aP |.

Now let ∆ ⊂ Ω. We define the pointwise stabilizer and setwise stabilizer respectively

by

P(∆) = σ | aσ = a, a ∈ ∆ and σ ∈ P




and

P∆ = σ | ∆σ = ∆, σ ∈ P .

It is clear that P(∆) and P∆ are subgroups of P. By definition, we know that

P(∆) = a∈∆

Pa,

and

P(∆1∪∆2) = P(∆1)

P(∆1) = (P(∆1))(∆2).

Applying Theorem 2.1.1, for a, b ∈ Ω we also know that

|P : Pa,b| = |aP ||bPa | = |bP ||aPb |.

Clearly, P(∆) ≤ P∆. Furthermore, we have the following result.

Theorem 2.1.2 P(∆) ⊳P∆.

Proof Let g ∈ P(∆) and h ∈ P∆. We prove that h−1gh ∈ P(∆). In fact, let a ∈ ∆, we

know that ah−1 ∈ ∆. Therefore,

ah−1gh = [(ah−1

)g]h = [ah−1

]h = a.

Whence, h−1gh ∈ P(∆).

2.1.3 Burnside Lemma. For counting the number of orbital sets Orb(Ω) of Ω under the

action of P, the following result, usually called Burnside Lemma is useful.

Theorem 2.1.3(Cauchy-Frobenius Lemma) Let P be a permutation group action on Ω.

Then

|Orb(Ω)| =1

|P |

x∈P

| fix( x)|,

where fix( x) = a ∈ Ω|a x = a.

Proof Define a set A = (a, x) ∈ Ω×P |a x = a. We count the number of elements of

A in two ways. Assuming the orbits of Ω

under the action of P areΩ

1,Ω

2, · · · ,Ω

|Orb(Ω)|.Applying Theorem 2.1.1(3), we get that

|A | =

|Orb(Ω)|i=1

a∈Ωi

Pa

=

|Orb(Ω)|i=1

a∈Ωi

|P ||Ωi| =

|Orb(Ω)|i=1

|P | = |Orb(Ω)||P |.



Sec.2.2 Transitive Groups 45

By definition, |A | =

x∈P|fix( x)|. Therefore,

|Orb(Ω)| =1

|P

| x∈P

|fix( x)|.


Notice that |fix( x)| remains constant on each conjugacy class of P , we get the fol-

lowing conclusion by Theorem 2.1.3.

Corollary 2.1.1 Let P be a permutation group action on Ω with conjugacy classes

C 1, C 2, · · · , C k . Then

|Orb(Ω)| =1

|P |k

i=1

|C i||fix( xi)|,

where xi ∈ C i.

Example 2.1.1 Let P = σ1, σ2, σ3, σ4, σ5, σ6σ7, σ8 be a permutation group action on

Ω = 1, 2, 3, 4, 5, 6, 7, 8, where

σ1 = 1P , σ2 = (1, 4, 3, 2)(5, 8, 7, 6),

σ3 = (1, 3)(2, 4)(5, 7)(6, 8), σ4 = (1, 2, 3, 4)(5, 6, 7, 8),

σ5 = (1, 7, 3, 5)(2, 6, 4, 8), σ6 = (1, 8, 3, 6)(2, 7, 4, 5),

σ7 = (1, 5, 3, 7)(2, 8, 4, 6), σ8 = (1, 6, 3, 8)(2, 5, 4, 7).

Calculation shows that

fix(1) = fix(2) = fix(3) = fix(4) = fix(5) = fix(6) = fix(7) = fix(8) = 1P,

Applying Theorem 2.1.3, the number of obits of Ω under the action of P is

|Orb(Ω)| =1

|P |

x∈P

|fix( x)| =1

8×

8i=1

1 = 1.

In fact, for ∀i ∈ Ω, the orbit of i under the action of P is

i

P

= 1, 2, 3, 4, 5, 6, 7, 8.

§2.2 TRANSITIVE GROUPS

2.2.1 Transitive Group. A permutation group P action on Ω is transitive if for x, y ∈ Ω,

there exists a permutation π ∈ P such that xπ = y. Whence, a transitive group P only has




one obit, i.e., Ω under the action of P . A permutation group P which is not transitive is

called intransitive. According to Theorem 2.1.1, we get the following result for transitive

groups.

Theorem 2.2.1 Let P be a transitive group acting on Ω , a ∈ Ω. Then |P | = |Ω||Pa| ,i.e., |P : Pa| = |Ω|.

A permutation group P action on Ω is said to be semi-regular if Pa = 1P for

∀a ∈ Ω. Furthermore, if P is transitive, Such a semi-regular group is called regular .

Corollary 2.2.1 Let P be a regular group action on Ω. Then |P | = |Ω|.Particularly, we know the following result for Abelian transitive groups.

Theorem 2.2.2 Let P be a transitive group action on Ω. If it is Abelian group, it must be regular.

Proof Let a ∈ Ω and π ∈ P. Then (Pa)π = Paπ by Theorem 2.1.1(2). But Pa⊳P

because P is Abelian. We know that Pa = Paπ for ∀π ∈ P . By assumption, P is

transitive. It follows that if aπ = a, then bπ = b for ∀b ∈ Ω. Thus Pa = 1P .

2.2.2 Multiply Transitive Group. Let P be a permutation group acting on Ω =

a1, a2, · · · , an and

Ωk =

(a1, a2,

· · ·, ak )

|ai

∈Ω, 1

≤i

≤k

.

Define P act on Ωk by

(a1, a2, · · · , ak )π = (aπ

1, aπ2, · · · , aπ

k ), π ∈ P.

If P acts transitive on Ωk , then P is said to be k-transitive on Ω. The following result is

a criterion on multiply transitive groups.

Theorem 2.2.3 For an integer k > 1 , a transitive permutation group P acting on Ω is

k-transitive if and only if for a fixed element a ∈ Ω , Pa is (k − 1)-transitive on Ω \ a.

Proof Assume that P is k -transitive acting on Ω and

(a1, a2, · · · , ak −1), (b1, b2, · · · , bk −1) ∈ Ω \ a.

Then ai a bi for 1 ≤ i ≤ k − 1. Notice that P is k -transitive. There is a permutation

π such that

(a1, a2, · · · , ak −1, a)π = (b1, b2, · · · , bk −1, a).



Sec.2.2 Transitive Groups 47

Thus π fixes a and maps (a1, a2, · · · , ak −1) to (b1, b2, · · · , bk −1), which shows that Pa acts

(k − 1)-transitively on Ω \ a.

Conversely, let Pa is (k −1)-transitive on Ω\a, (a1, a2, · · · , ak ), (b1, b2, · · · , bk ) ∈ Ωk .

By the transitivity of P acting on Ω, there exist elements π, π′ ∈ P such that aπ1 = a and

bπ′1 = a. Because Pa is (k − 1)-transitive on Ω \ a, there is an element σ ∈ Pa such that

((aπ2)σ, · · · , (aπ

k )σ) = (bπ′−1

2 , · · · , bπ′−1

k ).

Whence, aπσi = bπ′−1

i , i.e., aπσπ′i = bi for 2 ≤ i ≤ k . Since σ ∈ Pa, we know that aπσπ′

1 =

aσπ′= aπ′

= b1. Therefore, the element πσπ′ maps (a1, a2, · · · , ak ) to (b1, b2, · · · , bk ).

A simple calculation shows that

|Ωk

| = n(n − 1) · · · (n − k + 1).

Applying Theorems 2.2.1 and 2.2.3, we get the next conclusion.

Theorem 2.2.4 Let P be k-transitive on Ω. Then

n(n − 1) · · · (n − k + 1)||P |.

2.2.3 Sharply k -Transitive Group. A transitive group P on Ω is said to be sharply

k-transitive if P acts regularly on Ωk , i.e., for two k -tuples in Ωk , there is a unique permu-

tation in P mapping one k -tuple to another. The following is an immediately conclusion

by Theorem 2.1.1.

Theorem 2.2.5 A k-transitive group P acting on Ω with |Ω| = n is sharply k-transitive

if and only if |P | = n(n − 1) · · · (n − k + 1).

These symmetric and alternating groups are examples of multiply transitive groups

shown in the following.

Theorem 2.2.6 Let n

≥1 be an integer and Ω =

1, 2,

· · ·, n

. Then

(1) S Ω is sharply n-transitive;

(2) If n ≥ 3 , the alternating group AΩ is sharply (n − 2)-transitive group of degree n.

Proof For the claim (1), it is obvious by definition. We prove the claim (2). First, it

is easy to find that AΩ is transitive. Notice that if Ω = 1, 2, 3, AΩ is generated by (1, 2, 3).

It is regular and therefore sharply 1-transitive. Whence, the claim is true for n = 3. Now




assume this claim is true for all integers< n. Let n ≥ 4 and define H to be the stabilizer

of n. Then H acts on the set Ω \ n, produce all even permutations. By induction, H is

(n − 3)-transitive group on Ω \ n. Applying Theorem 2.2.3, AΩ is (n − 2)-transitive. Thus

| AΩ| =12 (n!) = n(n − 1) · · · 3. By Theorem 2.2.5, it is sharply (n − 2)-transitive.

More sharply multiply transitive groups are shown following. The reader is referred

to references [DiM1] and [Rob1] for their proofs.

Sharply 2, 3-transitive group. Let F be a Galois field GF (q) with q = pm for a prime

number p. Define X = F ∪∞ and think it as the projective line consisting of q + 1 lines.

Let L(q) be the set of all functions f : X → X of the form

f ( x) =

ax + b

cx + d

for a, b, c, d ∈ F with ad − bc 0, where the symbol ∞ is subject to rulers x + ∞ =

∞, ∞/∞ = 1, etc. Then it is easily to verify that L(q) is a group under the functional

composition. Define H (q) to be the stabilizer of ∞ in L(q), which is consisting of all

functions x → ax + b, a 0. Then H (q) is sharply 2-transitive on GF (q) of degree q and

L(q) is sharply 3-transitive on F ∪∞ of degree q + 1.

Particularly, if c = d = 0, i.e., for a linear transformation a and a vector v ∈ F d , we

define the a ffine transformation

t a,v : F d → F d by t a,v : u → ua + v.

Then the set of all t a,v form the a ffine group AGLd (q) of dimensional d ≥ 1.

Sharply 4, 5-transitive group Let Ω = 1, 2, 3, · · · , 11, 12 and

ϕ = (4, 5, 6)(7, 8, 9)(10, 11, 12), χ = (4, 7, 10)(5, 8, 11)(6, 9, 12),

ψ = (5, 7, 6, 10)(8, 9, 12, 11), ω = (5, 8, 6, 12)(7, 11, 10, 9),

π1 = (1, 4)(7, 8)(9, 11)(10, 12), π2 = (1, 2)(7, 10)(8, 11)(9, 12);

π3 = (2, 3)(7, 12)(8, 10)(9, 11).

Define M 11 = ϕ,χ,ψ,ω,π1, π2, π3 and M 12 = ϕ,χ,ψ,ω,π1, π2, called Mathieu groups.

Then M 11 is sharply 5-transitive of degree 12 with order 95040, and M 12 is sharply 4-

transitive of degree 11 on Ω \ 3 with order 7920.



Sec.2.3 Automorphisms of Groups 49

Theorem 2.2.7(Jordan) For an integer k ≥ 4 , let P be a sharply k-transitive group of

degree n which is neither symmetric nor alternating groups. Then either k = 4 and n = 11 ,

or k = 5 and n = 12.

Combining Examples 2.2.1, 2.2.2 with Theorem 2.2.7, we know that there are sharply

k -transitive group of finite degree if and only if 1 ≤ k ≤ 5.

§2.3 AUTOMORPHISMS OF GROUPS

2.3.1 Automorphism Group. An automorphism of a group (G ; ) is an isomorphism

from G to G . All automorphisms of a group form a group under the functional compo-

sition, i.e., θς ( x) = θ (ς ( x)) for x ∈ G . Denoted by AutG , which is a permutation group

action on G itself. We discuss this kind of permutation groups in this section.

Example 2.3.1 Let G = e, a, b, c be an Abelian 4-group with operation · determined by

the following table.

· e a b c

e e a b c

a a e c b

b b c e a

c c b a e

Table 2.3.1

We determine the automorphism group AutG . Notice that e is the identity element of

G . By property (H1) of homomorphism, if θ is an automorphism on G , then θ (e) = e.

Whence, there are six cases for possible θ following:

θ 1 =

e a b c

e a b c

, θ 2 =

e a b c

e a c b

,

θ 3 =

e a b c

e b a c

, θ 4 =

e a b c

e b c a

,




θ 5 =

e a b c

e c a b

, θ 6 =

e a b c

e c b a

.

It is easily to check that all these θ i, 1 ≤ i ≤ 6 are automorphisms of (G ; ·). We get the

automorphism group

AutG = θ 1, θ 2, θ 3, θ 4, θ 5, θ 6.

Let x, g ∈ G . An element xg = g−1 x g is called the conjugate of x by g. Define a

mapping gτ : G → G by gτ( x) = xg. Then ( x y)g = xg yg and gτ(g−1)τ = 1AutG = (g−1)τgτ.

So gτ ∈ AutG , i.e., an automorphism on (G ; ). Such an automorphism gτ is called

the inner automorphism of (G ; ) induced by g. It is easily to check that all such inner

automorphisms form a subgroup of AutG , denoted by InnG .

Theorem 2.3.1 Let (G ; ) be a group. Then the mapping τ : G → AutG defined by

τ( x) = gτ( x) = xg for ∀ x ∈ G is a homomorphism with image InnG and kernel the set of

elements commutating with every element of G .

Proof By definition, we know that x(gh)τ

= (gh)−1 x(gh) = h−1 g−1 xgh =

( xgτ

)hτ

. So (g h)τ = gτhτ, which means that τ is a homomorphism.

Notice that gτ = 1AutG is equivalent to g−1 xg = x by definition. Namely, g x = xg

for

∀ x

∈G . This completes the proof.

Definition 2.3.1 The center Z (G ) of a group (G ; ) is defined by

Z (G ) = x ∈ G | x g = g x for all g ∈ G .

Then Theorem 2.3.1 can be restated as follows.

Theorem 2.3.2 Let (G ; ) be a group. Then Z (G )⊳ G and G / Z (G ) ≃ InnG .

The properties of inner automorphism group InnG induced it to be a normal sub-

group of AutG

following.

Theorem 2.3.3 Let (G ; ) be a group. Then InnG ⊳AutG .

Proof Let g ∈ G and h ∈ AutG . Then for ∀ x ∈ G ,

hgτh−1( x) = hgτ(h−1( x)) = h(g−1 h−1( x) g)

= h−1(g) x h(g) = xh(g) ∈ InnG .






A relation between the center and normalizer of subgroup of a group is determined

in the next result.

Theorem 2.3.6 Let (G ;

) be a group, H

≤G . Then Z (H )⊳ N G (H ).

Proof If g ∈ N G (H ), let gτ denote the mapping h → g−1 h h. It is clear an

automorphism of H . Furthermore, τ : N G (H ) → AutH is a homomorphism with

kernel Z (H ). Then this result follows from Theorem 1.3.3.

2.3.2 Characteristic Subgroup. Let (G ; ) be a group, H ≤ G and g ∈ AutG . By

definition, there must be g(H ) ≃ H but g(H ) H in general. If g(H ) = H for

∀g ∈ AutG , then such a subgroup is particular and called a characteristic subgroup of

(G ; ). For example, the center of a group is in fact a characteristic subgroup by Definition

2.3.1.

According to the definition of normal subgroup, For ∀h ∈ InnG , a subgroup H of

a group (G ; ) is norma if and only if h(H ) = H for ∀h ∈ InnG . So a characteristic

subgroup must be a normal subgroup. But the converse is not always true.

Example 2.3.2 Let D 8 = e, a, a2, a3, b, b · a, b · a2, b · a3 be a dihedral group of order 8

with an operation · determined by the following table.

· e a a2 a3 b a · b a2 · b a3 · b

e e a a2 a3 b a · b a2 · b a3 · b

a a a2 a3 e a · b a2 · b a3 · b b

a2 a2 a3 e a a2 · b a3 · b b a · b

a3 a3 e a a2 a3 · b b a · b a2 · b

b b a3 · b a2 · b a · b a2 a e a3

a · b a · b b a3 · b a2 · b a3 a2 a e

a2 · b a2 · b a · b b a3 · b e a3 a2 a

a3 · b a3 · b a2 · b a · b b a e a3 a2

Table 2.3.2

Notice that all subgroups of D 8 are normal and a is a unique element of degree 2. So

(

a2

; ) is a characteristic subgroup of D 8.

Now let b = e, b, a2, a2 · b. Clearly, it is a subgroup of D 8. We prove it is not a

characteristic subgroup of D 8. In fact, let φ : D → D be a one-to-one mapping defined by



Sec.2.3 Automorphisms of Groups 53

e → e, a → a, a2 → a2, a3 → a3,

b → a · b, a · b → a2 · b, a2 · b → a3 · b, a3 · b → b.

Then φ is an automorphism. But

φ(b) = e, a · b, a2, a3 · b b .

Therefore, it is not a characteristic subgroup of D 8.

The following result is clear by definition.

Theorem 2.3.7 If G 1 ≤ G is a characteristic subgroup of G and G 2 ≤ G 1 a characteristic

subgroup of G 1 , then G 2 is also a characteristic subgroup of G .

2.3.3 Commutator Subgroup. Let (G ;

) be a group and a, b∈

G . The element

[a, b] = a−1 b−1 a b

is called the commutator of a and b. Obviously, a group (G ; ) is commutative if and only

if [a, b] = 1G for ∀a, b ∈ G . The commutator subgroup is generated by all commutators

of (G ; ), denoted by G ′ or [G , G ], i.e.,

G ′ = [a, b] | a, b ∈ G .

Theorem 2.3.8 [S n, S n] = An.

Proof Notice that we can always represent a permutation by product of involutions.

By the definition of commutator, it is obvious that [S n, S n] ⊂ An. Now for ∀g ∈ An we can

always write it as g = (as11, as21)(as12, as22) · · · (as1m, as2m) with m ≡ 0(mod2) by definition,

where asi j ∈ 1, 2, · · · , n for i = 1, 2 and 1 ≤ j ≤ m. Calculation shows that

(i, j)( j, k ) = ( j, k )(i, j)( j, k )(i, j) = [( j, k ), (i, j)]

if i j, j k and

(i, j)(k , l) = (i, j)( j, k )( j, k )(k , l) = [( j, k ), (i, j)][(k , l), ( j, k )]

if i, j, k , l are all distinct. Whence, each element in An can be written as a product of

elements in [S n, S n], i.e., An ⊂ [S n, S n].

A commutator subgroup is always a characteristic subgroup, such as those shown in

the next result.




Theorem 2.3.9 Any commutator subgroup of a group (G ; ) is a characteristic subgroup.

Proof Let φ ∈ G . We prove φ(G ′) = G ′. In fact, for ∀a, b ∈ G , we know that

φ([a, b]) = φ(a−1

b−1

a b)

= φ(a−1) φ(b−1) φ(a) φ(b)

= φ−1(a) φ−1(b) φ(a) φ(b) = [φ(a), φ(b)].

Whence, G ′ is a characteristic subgroup of (G ; ).

Corollary 2.3.1 Any non-commutative group (G ; ) has a non-trivial characteristic sub-

group.

Proof If (G ;

) is non-commutative, then there are elements a, b∈

G such that

[a, b] 1G . Whence, it has a non-trivial characteristic subgroup G ′ at least.

The most important properties of commutator subgroups is the next.

Theorem 2.3.10 Let (G ; ) be a group. Then

(1) The quotient group G /G ′ is commutative;

(2) The quotient group G /H is commutative for H ⊳ G if and only if H ≥ G ′.

Proof (1) Let a, b ∈ G . Then

(a G ′)−1 (b G ′)−1 (a G ′) (b G ′)

= a−1 G ′ b−1 G ′ a G ′ b G ′

= (a−1 b−1 a b) G ′ = G ′.

Therefore, a G ′ b G ′ = b G ′ a G ′.

(2) Notice that G /H is commutative if and only if for a, b ∈ G ,

a H b H = b H a H .

This equality is equivalent to

(a H )−1 (b H )−1 (a H ) (b H ) = H ,

i.e., (a−1 b−1 a b) H = H . Whence, we find that [a, b] = a−1 b−1 a b ∈ H ,

which means that H ≥ G ′.



Sec.2.4 P-Groups 55

§2.4 P-GROUPS

As one applying fields of permutations to abstract groups, we discuss p-groups in this

section.

2.4.1 Sylow Theorem. By definition, a Sylow p-subgroup of a group (G , ) with |G | =

pαn, ( p, n) = 1 is essentially a subgroup with maximum order pα. Such p-subgroups are

important for knowing the structure of finite groups, for example, the structure Theorems

1.4.4-1.4.6 for Abelian groups.

Theorem 2.4.1(Sylow’s First Theorem) Let (G ; ) be a finite group, p a prime number

and |G | = pαn, ( p, n) = 1. Then for any integer i, 1 ≤ i ≤ α , there exists a subgroup of

order pi , particularly, the Sylow subgroup always exists.

Proof The proof is by induction on |G |. Clearly, our conclusion is true for n = 1.

Assume it is true for all groups of order≤ pαn.

Denoted by z the order of center Z (G ). Notice that Z (G ) is a Abelian subgroup

of G . If p| z, there exists an element a of order p by Theorem 1.4.6. So a is a normal

subgroup of G with order p. We get a quotient group G / a with order pα−1n < n.

By the induction assumption, we know that there are subgroups Pi/ a of order pi, i =

1, 2, · · · , α − 1 in G / a. So Pi, i = 1, 2, · · · , α − 1 are subgroups of order pi+1 in G .

Now if p | z, let C 1, C 2, · · · , C s be conjugacy classes of G . Notice that p||G | but p | z.By

|G | = |Z (G )| +

si=1

|C i|,

we know that there must be an integer l, 1 ≤ i ≤ s such that p ||C l|. Let b ∈ C l. Then

N G (b) = g ∈ G |g−1 b g = b

is a subgroup of G with index

|G : Z G (b)| = hl > 1.

Since pα and Z G (b) < pαn, by the induction assumption we know that there are subgroups

of order pi for 1 ≤ i ≤ α in Z G (b) ≤ G .

Corollary 2.4.1 Let (G ; ) be a finite group and p a prime number. If p||G | , then there

are elements of order p in (G ; ).




Theorem 2.4.2(Sylow’s Second Theorem) Let (G ; ) be a finite group, p a prime with

p||G |. Then

(1) If n p is the number of Sylow p-subgroups in G , then n p

≡1(mod p);

(2) All Sylow subgroups are conjugate in (G ; ).

Proof Let P, P1, P2, · · · , Pr be all Sylow p-subgroups in G . Notice that a conjugacy

subgroup of Sylow p-subgroup is still a Sylow subgroup of G . For ∀a ∈ G , define a

permutation

σa =

P P1 · · · Pr

a−1 P a a−1 P1 a · · · a−1 Pr a

.

and S p = σa|a ∈ P. Then S p is a homomorphic image of P. It is also a p-subgroup.

If Pk is invariant under the action S p for an integer 1 ≤ k ≤ r , then a Pk = Pk a for

∀a ∈ P. Whence, PPk is a p-subgroup of G . But P, Pk are Sylow p-subgroups of G . We

get PPk = P = Pk , contradicts to the assumption. So all Pk , 1 ≤ k ≤ r are not invariant

under the action of S p except P. By Theorem 2.1.1, we know that |PS p

k |||S p| for 1 ≤ k ≤ r .

Let PS p

k 1, P

S p

k 2, · · · , P

S p

k t be a partition of P1, P2, · · · , Pr . Then

n p = 1 + r = 1 +

t i=1

|PS p

k i| ≡ 1(mod p).

This is the conclusion (1).For the conclusion (2), assume there are s conjugate subgroups to P. Similarly, we

know that s ≡ 1(mod p). If there exists another conjugcy class in which there are s1

Sylow p-subgroups, we can also find s1 ≡ 1(mod p), a contradiction. So there are just

one conjugate class of Sylow p-subgroups. This fact enables us to know that all Sylow

subgroups are conjugate in (G ; ).

Corollary 2.4.2 Let P be a Sylow p-subgroup of (G ; ). Then

(1) P⊳ G if and only if P is uniquely the Sylow p-subgroup of (G ;

);

(2) P is uniquely the Sylow p-subgroup of N G (P).

Theorem 2.4.3(Sylow’s Third Theorem) Let (G ; ) be a finite group, p a prime with

p||G |. Then each p-subgroup A is a subgroup of a Sylow p-subgroup of (G ; ).

Proof Let σa be the same in the proof of Theorem 2.4.2 and S A = σa|a ∈ A.

Consider the action of S A on Sylow p-subgroups P, P1, · · · , Pr . Similar to the proof of



Sec.2.4 P-Groups 57

Theorem 2.4.2(1), we know that |PS A

k |||S A| for 1 ≤ k ≤ r . Because of r ≡ 0(mod p).

Whence, there are at least one obit with only one Sylow p-subgroups. Let it be Pl. Then

for ∀a ∈ A, a−1 Pl a = Pl. So APl is a p-subgroup. Notice that Pl ≤ APl. We get that

APl = Pl, i.e., A ≤ Pl.

2.4.2 Application of Sylow Theorem. Sylow theorems enables one to know the p-

subgroup structures of finite groups.

Theorem 2.4.4 Let P be a Sylow p-subgroup of (G ; ). Then

(1) If N G (P) ≤ H ≤ G , then H = N G (H );

(2) If N ⊳ G , then P ∩ N is a sylow p-subgroup of ( N ; ) and PN / N is a Sylow

p-subgroup of (G/ N ; ).

Proof (1) Let x ∈ N G (H ). Because P ≤ H ⊳ N G (H ), we know that x−1 P x ≤ H .

Clearly, P and x−1 P x are both Sylow p-subgroup of H . By Theorem 2.4.2, there is

an element h ∈ H such that x−1 P x = h−1 P h. Whence, x h−1 ∈ N G (P) ≤ H .

So x ∈ H , i.e., H = N G (H ).

(2) Notice that PN is a union of cosets a P, a ∈ N and N a union of cosets b (P ∩ N ), b ∈ N . Now let a, b ∈ N . By

a P = b P ⇔ a−1 b ∈ P ⇔ a−1 b ∈ N ∩ P ⇔ a N ∩ P = b N ∩ P,

we get that | N : P ∩ N | = |PN : P|, which is prime to p. Since N ∩ P, NP/ N are respective

p-subgroups of N or G / N by Theorem 1.2.6, this relation implies that they must be Sylow

p-subgroup of N or G / N .

Theorem 2.4.5(Fratini) Let N ⊳ G and P a Sylow p-subgroup of ( N ; ). Then G =

N G (P) N.

Proof Choose a ∈ G . Since N ⊳ G , we know that a−1 P a ≤ N , which implies

that a−1

P

a is also a Sylow p-subgroup of ( N ;

). According to Theorem 2.4.2, there

is b ∈ N such that b−1 (a−1 P a) b = P. Whence, a b ∈ N G (P), i.e., a ∈ N G (P) N .

Thus G = N G (P) N .

As we known, a finite group with prime power pα for an integer α is called a p-group

in group theory. For p-groups, we know the following results.

Theorem 2.4.6 Let (G ; ) be a non-trivial p-group. Then Z (G ) > 1G .




Proof Let |G | = pm, m an integer and C 1 = 1G , C 2, · · · , C s conjugate classes of G .

Bys

i=1

|C i| = |G | = pm,

we know that |C i| = 1 or a multiple of p by Theorem 2.4.5. But |C 1| = 1. Whence, there

are at least an integer k , 1 ≤ k ≤ s such that |C k | = 1, i.e., C k = a. Then a ∈ Z (G ).

Theorem 2.4.7 Let p be a prime number. A group (G ; ) of order p or p2 is Abelian.

Proof If |G | = p, then G = a with a p = 1G by Theorem 1.2.6.

Now let |G | = p2. If there is an element b ∈ G with o(b) = p2, then G = b, a cyclic

group of order p2 by Theorem 1.2.6. If such b does not exist, by Theorem 2.4.6 Z (G ) >

1G

, we can always choose 1G a

∈Z (G ) and b

∈G

\ Z (G ). Then o(a) = o(b) = p by

Theorem 1.2.6. We get that Z (G ) = a and G / Z (G ) = b Z (G ). Whence, G = a, bwith a b = b a and o(a) = o(b) = p. So it is Abelian.

For groups of order pq or p2q, we have the following result.

Theorem 2.4.8 Let p, q be odd prime numbers, p q. Then groups (G ; ) of order pq or

p2q are not simple groups.

Proof By Sylow’s theorem, we know that there are n p ≡ 1(mod p) Sylow q-subgroups

P in (G ; ). Let n p = 1 + kp for an integer k .

If |G | = pq, p ≥ q, we get that p(1 + kp)| pq, i.e., 1 + kp|q. So k = 0 and there is onlyone p-subgroup P in (G ; ). We know that P ⊳ G . Similarly, if p ≤ q, then the Sylow

q-subgroup Q⊳ G . So a group of order pq is not simple.

If |G | = p2q and p ≥ q, then 1 + kp|q implies that k = 0, and the only one p-subgroup

P⊳ G . Otherwise, p ≤ q, we know 1 + lq| p2. Notice that p ≤ q, we know that nq = 1 or

p2. But if nq = p2, i.e., lq = p2 − 1, we get that q|( p − 1)( p + 1). Whence, q = p + 1. It

is impossible since p and p + 1 can not both be prime numbers. So nq = 1. Let Q be the

only one Sylow q-subgroup in (G ; ). Then Q⊳G . Therefore, a group of order p2q is not

simple.

2.4.3 Listing p-Group. For listing p-groups, we need a symbol

λ

p

, i.e., the Legendre

symbol in number theory. For a prime p |λ, the number

λ p

is defined by

λ

p

=

1, if x2 ≡ λ(mod p) has solution;

−1, if x2 ≡ λ(mod p) has no solution.



Sec.2.4 P-Groups 59

We have known that λ

p

≡ λ

p−12 (mod p)

and the well-known Gauss reciprocity lawq

p

pq

= (−1)

( p−1)(q−1)4

for prime numbers p and q in number theory, .

Completely list all p-groups is a very difficult work. Today, we can only list those of

p-groups with small power. For example, these p-groups of orders pn for 1 ≤ n ≤ 4 are

listed in Tables 2.4.1 − 2.4.4 without proofs.

|G | p-group Abelian?

p (1) a, a p

= 1G Yes p2 (1) a, a p2

= 1G Yes

(2) a, b, a p = b p = 1G , a b = b a Yes

(1) a, a p3

= 1G Yes

(2) a, b, a p2

= b p = 1G , a b = b a Yes

(3) a, b, c, a p = b p = c p = 1G , a b = b a,

a c = c a, b c = c b Yes

p3 (4) a, b, a p2

= b p = 1G , b−1 a b = a1+ p No

( p 2) (5) a, b, c, a p

= b p

= c p

= 1G ,a b = b a c,c a = a c, c b = b c No

Table 2.4.1

For p = 2, these 2-groups of order 23 are completely listed in Table 2.4.2.

|G | 2-group Abelian?

(1) a, a8 = 1G Yes

(2)

a, b

, a4 = b2 = 1G , a

b = b

a Yes

23 (3) a, b, c, a2 = b2 = c2 = 1G , a b = b a,

a c = c a, b c = c b Yes

(4) Q8 = a, b, a4 = 1G , b2 = a2 b−1 a b = a−1 No

(5) D8 = a, b, a4 = b2 = 1G , b−1 a b = a−1 No

Table 2.4.2





Sec.2.5 Primitive Groups 61

23 in Table 2.4.2. Similarly, these non-Abelian 2-groups of order 24 are listed in Table

2.4.4 following.

|G

|2-group Abelian?

(1) a, b, a8 = b2 = 1G , b−1 a b = a−1 No

(2) a, b, a8 = b2 = 1G , b−1 a b = a3 No

(3) a, b, a8 = b2 = 1G , b−1 a b = a5 No

(4) a, b, a8 = 1G , b2 = a4, b−1 a b = a−1 No

24 (5) a, b, a4 = b4 = 1G , b−1 a b = a−1 No

(6) a, b, c, a4 = b2 = c2 = 1G , b−1 a b = a,

c−1 a c = a, [b, c] = a2 No

(7)

a, b, c

, a4 = b2 = c2 = 1G , b−1

a

b = a,

c−1 a c = a−1, [b, c] = a2 No

(8) a, b, c, a4 = b2 = 1G , c2 = a2, b−1 a b = a,

c−1 a c = a−1, [b, c] = 1G No

(9) a, b, c, a4 = b2 = c2 = 1G , b−1 a b = a,

c−1 a c = a b, [b, c] = 1G No

Table 2.4.4

A complete proof for listing results in Tables 2.4.1-2.4.4 can be found in references,for example, [Zha1] or [Xum1].

§2.5 PRIMITIVE GROUPS

2.5.1 Imprimitive Block. Let P be a permutation group action on Ω. A proper subset

A ⊂ Ω, | A| ≥ 2 is called an imprimitive block of P if for ∀π ∈ P , either A = Aπ

or A∩

Aπ =∅

. If such blocks A exist, we sayP

imprimitive. Otherwise, it is called

primitive, i.e., P has no imprimitive blocks.

Example 2.5.1 Let P be a permutation group generated by

g = (1, 2, 3, 4, 5, 6) and h = (2, 6)(3, 5).

Notice that P is transitive on Ω = 1, 2, 3, 4, 5, 6 and hg = g5h. There are only 12




elements with form glhm, where l = 0, 1, 2, 3, 4, 5 and m = 0, 1. Let A = 1, 4. Then

1, 4g = 2, 5, 1, 4g2

= 3, 6,

1, 4g

3

= 1, 4, 1, 4h = 1, 4.

Whence, Aτ = A or Aτ ∩ A = ∅ for ∀τ ∈ P, i.e., A is an imprimitive block.

The following result is followed immediately by Theorem 2.1.1 on primitive groups.

Theorem 2.5.1 Let P be a transitive group actin on Ω , A an imprimitive block of P and

H the subgroup of all π in P such that Aπ = A. Then

(1) The subsets Aτ, τ ∈ P : H form a partition of Ω;

(2) |Ω| = | A||P : H |.Proof Let a ∈ Ω and b ∈ A. By the transitivity of P on Ω, there is a permutation

π ∈ P such that a = bπ. Writing π = στ with σ ∈ H and τ ∈ P : H , we find that

a = (bσ)τ ∈ Aτ. Whence, Ω is certainly the union of Aτ, τ ∈ H . Now if Aτ ∩ Aτ′ ∅, then

A ∩ ( Aτ′)τ−1

∅. Consequently, A = ( Aτ′)τ−1

and τ′τ−1 ∈ H . But τ, τ′ ∈ P : H , we get that

τ = τ′. So Aτ, τ ∈ P : H is a partition of Ω. Thus we establish (1).

Notice that | A| = | Aτ| for τ ∈ P : H . We immediately get that |Ω| = | A||P : H | by

(1).

2.5.2 Primitive Group. Applying Theorem 2.3.1, the following result on primitive

groups is obvious.

Theorem 2.5.2 A transitive group of prime degree is primitive.

These multiply at least 2-transitive groups constitute a frequently encountered prim-

itive groups shown following.

Theorem 2.5.3 Every 2-transitive group is primitive.

Proof Let P be a 2-transitive group action on Ω. If it is imprimitive, then there

exists an imprimitive block A of P. Whence we can find elements a, b ∈ A and c ∈ Ω\ A.By the 2-transitivity, there is an element π ∈ P such that (a, b)π = (a, c). So a ∈ A ∩ Aπ.

Consequently, A = Aπ. But this will implies that c = bπ ∈ A, a contradiction.

Let (G ; ) be a group. A subgroup H < G is maximal if there are no subgroups

K < G such that H < K < G . The next result is a more valuable criterion on primitiv-

ity of permutation groups.




Theorem 2.5.4 A transitive group P action on Ω is primitive if and only if Pa is maximal

for ∀a ∈ Ω.

Proof If Pa is not maximal, then there exists a subgroup H of P such that Pa <

H < P . Define a subset of Ω by

A = aτ|τ ∈ H .

Then | A| ≥ 2 because of H > Pa. First, if A = Ω, then for ∀π ∈ P we can find

an element σ ∈ H such that aπ = aσ. Thus πσ−1 ∈ Pa, which gives π ∈ H and

H = P . Now if there is π ∈ P with A ∩ Aπ ∅ hold, then there are σ1, σ2 ∈ H such

that aσ1 = aσ2π. Thus σ−11 ∈ Pa < H . Whence, π ∈ H , which implies that A = Aπ.

Therefore, A is an iprimitive block and P is imprimitive.Conversely, let A be an imprimitive block of P . By the transitivity of P on Ω, we

can assume that a ∈ A. Define

H = π ∈ P | Aπ = A, π ∈ P.

Then H ≤ G . For b, c ∈ A, there is a π ∈ G such that bπ = c. Thus c ∈ A ∩ Aπ. Whence,

A = Aπ and π ∈ H by definition. Therefore, H is transitive on A. Consequently,

A =|H : H a

|. Now if π

∈P , then a = aπ

∈A

∩ Aπ. So A = Aπ and π

∈H . Thereafter,

Pa < H and Pa = H a. Applying Theorem 2.1.1, we know that |Ω| = |P : Pa| and

| A| = |H : H a| = |H : Pa|. So Pa < H < P and Pa is not maximal in P .

Corollary 2.5.1 Let P be a transitive group action on Ω. If there is a proper subset

A ⊂ Ω , | A| ≥ 2 such that

a ∈ A, aπ ∈ A ⇒ Aπ= A

for π ∈ P , then P is imprimitive.

Proof By Theorem 2.5.4, we only need to prove that Pa < P A < P, i.e., Pa is

not maximal of P . In fact, Pa ≤ P A is obvious by definition. Applying the transitivity

of P , for ∀b ∈ A there is an element σ ∈ P such that aσ = b. Clearly, σ ∈ P A, but

σ Pa. Whence, Pa < P A.

Now let c ∈ Ω \ A. Applying the transitivity of P again, there is an element τ ∈ P




such that aτ = c. Clearly, τ ∈ G but τ G A. So we finally get that

Pa < P A < P ,

i.e., Pa is not maximal in P .

Theorem 2.5.5 Let P be a nontrivial primitive group action on Ω. If N ⊳P , then N

is transitive on Ω.

Proof Let a ∈ Ω and A = aτ|τ ∈ N . Notice that (aσ)π = (aπ)σπ

and σπ ∈ N if

π ∈ P , σ ∈ N . Thus Aπ is an obit containing aπ. Whence, A = Aπ or a ∩ Aπ = ∅, which

implies that A is an imprimitive block. This is impossible because P is primitive on Ω.

Whence, A = Ω, i.e., N is transitive on Ω.

Theorem 2.5.5 also implies the next result for imprimitive groups.

Corollary 2.5.2 Let P be a transitive group action on Ω with a non-transitive normal

subgroup N . Then P is imprimitive.

The following result relates primitive groups with simple groups.

Theorem 2.5.6 Let P be a nontrivial primitive group action on Ω. If there is an element

x ∈ Ω such that P x is simple, then there is a subgroup N ⊳P action regularly on Ω

unless P is itself simple.

Proof If P is not simple, then there is a proper normal subgroup N ⊳P. Consider

N ∩ P x, which is a normal subgroup of P x. Notice that P x is simple. We know that

N ∩ P x = P x or 1P.

Now if N ∩P x = P x, then P x ≤ N . Applying Theorem 2.5.5, we know that N is

transitive on Ω. Whence, N < P x since xς = x for ∀ς ∈ P x, i.e., P x is not transitive on

Ω. By Theorem 2.5.4, there must be N = P, a contradiction. Whence, N ∩P x = 1P.

Applying the transitivity of N on Ω, we immediately get that N y = 1P for ∀ y ∈ Ω, i.e.,

N acts regularly on Ω.

2.5.3 Regular Normal Subgroup. Theorem 2.5.5 shows the importance of normal

subgroups of primitive groups. In fact, we can determine all regular normal subgroups of

multiply transitive groups. First, we prove the next result.

Theorem 2.5.7 Let (G ; ) be a nontrivial finite group and P = AutG .




(1) If P is transitive, then (G ; ) is an elementary Abelian p-group for some prime

p;

(2) If P is 2-transitive, then either p = 2 or |G | = 3;

(3) If P is 3-transitive, then |G | = 4;

(4) P can not be 4-transitive.

Proof (1) Let p be a prime dividing |G |. Then there exists an element x of order

p by Corollary 2.4.1. By the transitivity we know that every element in G \ 1G is the

form xτ, τ ∈ P and hence of order p also. Thus G is a finite p-group and its center

Z (G ) is nontrivial by Theorem 2.4.6. By definition, Z (G ) is characteristic in (G ; ) and

thus is invariant in G . Applying the transitivity of P enables us to know that Z (G ) = G .

Whence, G is an elementary Abelian p-groups.

(2) If p > 2, let x ∈ G with x 1G . Thus x x−1. If there is also an element y ∈ G ,

y 1G , x, x−1, then the 2-transitivity assures us of a τ ∈ P such that ( x, x−1)τ = ( x, y).

Plainly, this fact implies that y = x−1, a contradiction. Therefore, G = 1G , x, x−1 and

|G | = 3.

(3) If P is 3-transitive on G \1G , the later must has 3 elements at least, i.e., |G | ≥ 4.

Applying (2) we know that G is an elementary Abelian 2-group. Let H = 1, x, y, x ybe a subgroup of order 4. If there is an element z ∈ G \ H , then x z, y z and x y z

are distinct. So there must be an automorphism τ ∈ P such that

xτ = x z, yτ = y z and ( x y)τ = x y z

by the 3-transitivity of P on G . However, these relations imply that z = 1G , a contradic-

tion. Whence, H = G .

(4) If P were 4-transitive, it would be 3-transitive and |G | = 4 by (3), which excludes

the possibility of 4-transitivity. Whence, P can not be 4-transitive.

By Theorem 2.5.7, the regular normal subgroups of multiply transitive groups can

be completely determined.

Theorem 2.5.8 Let P be a k-transitive group of degree n with k ≥ 2 and N a nontrivial

regular normal subgroup of P. Then,

(1) If k = 2 , then n = |N | = pm and N is an elementary Abelian p-group for some

prime p and integer m;

(2) If k = 3 , then either p = 2 or n = 3;




(3) If k = 4 , then n = 4;

(4) k ≥ 5 is impossible.

Proof Clearly, 1 < k

≤n. Let P be a k -transitive group acting on Ω with

|Ω

|= n

and a ∈ Ω. By Theorem 2.2.3, we know that Pa is (k − 1)-transitive on Ω \ a.

Consider the action of Pa on N \ 1P by conjugation. Now if π ∈ N \ 1P, by

the regularity of N we know that aπ a. Thus there is a mapping Θ from N \ 1P to

Ω \ a determined by Θ : π → aπ. Applying the regularity of N again, we know that

Θ is injective. Besides, since N is transitive by Theorem 2.5.5, we know that Θ is also

surjective. Whence,

Θ : N \ 1P → Ω \ a

is a bijection.Now let 1P π ∈ P and σ ∈ Pa. Then we have that (aπ)σ = aπσ

, or (Θ(π))σ =

Θ(πσ). Thereafter, the permutation representations of Pa on N \ 1P and Ω \ a are

equivalent. Whence Pa is (k − 1)-transitive on N \ 1P. Notice that Pa ≤ AutN . We

therefore know that AutN is (k − 1)-transitive on N \ 1P also. By Theorem 2.5.7, we

immediately get all these conclusions (1) − (4).

2.5.4 O’Nan-Scott Theorem. The main approach in classification of primitive groups

is to study the subgroup generated by the minimal subgroups, i.e., the socle of a group

defined following.

Definition 2.5.1 Let (G ; ) be a group. A minimal normal subgroup of (G ; ) is such a

normal subgroup (N ; ) , N 1G which does not contain other properly nontrivial

normal subgroup of G .

Definition 2.5.2 Let (G ; ) be a group with all minimal normal subgroups N 1, N 2, · · · ,

N m. The socle soc(G ) of (G ; ) is determined by

soc(G ) =

N 1, N 2,

· · ·, N m

.

Then we know the following results on socle of finite groups without proofs.

Theorem 2.5.9 Let (G ; ) be a nontrivial finite group. Then

(1) If K is a minimal normal subgroup and L a normal subgroup of (G ; ) , then either

K ≤ L or K , L = K × L;




(2) There exist minimal normal subgroups K 1, K 2, · · · , K m of (G ; ) such that

soc(G ) = K 1 × K 2 × · · · × K m;

(3) Every minimal normal subgroup K of (G ; ) is a direct product K = T 1 × T 2 ×· · · × T k , where these T i, 1 ≤ i ≤ k are simple normal subgroups of K which are conjugate

under (G ; );

(4) If these subgroup K i, 1 ≤ i ≤ m in (2) are all non-Abelian, then K 1, K 2, · · · , K m

are the only minimal normal subgroups of (G ; ). Similarly, if these T i, 1 ≤ i ≤ k in (3)

are non-Abelian, then they are the only minimal normal subgroups of K.

Theorem 2.5.10 Let P be a finite primitive group of S Ω and K a minimal normal sub-

group of P. Then exactly one of the following holds:

(1) For some prime p and integer d, K is a regular elementary Abelian group of

order pd , and soc(P) = K = Z G (K ) , where Z G (K ) is the centralizer of K in P;

(2) K is a regular non-Abelian group, Z G (K ) is a minimal normal subgroup of P

which is permutation isomorphic to K, and soc(P) = K × Z G (K );

(3) K is non-Abelian, Z G (K ) = 1P and soc(P) = K.

Particularly, for the socle of a primitive group, we get the following conclusion.

Corollary 2.5.3 Let P be a finite primitive group of S Ω with the socle H. Then

(1) H is a direct product of isomorphic simple groups;

(2) H is a minimal normal subgroup of N S Ω( H ). Moreover, if H is not regular, then

it is the only minimal normal subgroup of N S Ω( H ).

Let Ω and ∆ be two sets or groups. Denoted by Fun(Ω, ∆) the set of all functions

from Ω into ∆. For two groups K , H acting on a non-empty set Ω, the wreath product

K wr Ω H of K by H with respect to this action is defined to be the semidirect product

Fun(Ω, K )⋊H , where H acts on the group Fun(Ω, K ) is determined by

f γ

(a) = f (aγ −1

) for all f ∈ Fun(Ω, K ), a ∈ Ω and γ ∈ H .

and the operation · in Fun(Ω, K ) × H is defined to be

( f 1, g1) · ( f 2, g2) = ( f 1 f g−1

1

2, g1g2).

Usually, the group B = ( f , 1H )| f ∈ Fun(Ω, K ) is called the base group of the wreath

product K wr Ω H .




A permutation group P acting on Ω with the socle H is said to be diagonal type

if P is a subgroup of the normalizer N S Ω( H ) such that P contains the base group H =

T 1 × T 2 × · · · × T m. Then by Theorem 2.5.9 these groups T 1, T 2, · · · , T m are the only

minimal normal subgroups of H and H ⊳ P . So P acts by conjugation on the setT 1, T 2, · · · , T m. Then we know the next result characterizing those primitive groups of

diagonal type without proof.

Theorem 2.5.11 Let P ≤ N S Ω( H ) be a diagonal type group with the socle H = T 1 ×

T 2 × · · · × T m. Then P is primitive subgroup of S Ω either if

(1) m = 2; or

(2) m ≥ 3 and the action of P by conjugation on T 1, T 2, · · · , T m of the minimal

normal subgroups of H is primitive.

Now we can present the O’Nan-Scott theorem following, which characterizes the

structure of primitive groups.

Theorem 2.5.12(O’Nan-Scott Theorem) Let P be a finite primitive group of degree n

and H the socle of P. Then either

(1) H is a regular elementary Abelian p-group for some prime p, n = pm = |H |and P is isomorphic to a subgroup of the a ffine group AGLm( p); or

(2) H is isomorphic to a direct power T m of a non-Abelian simple group T and one

of the following holds:

(i) m = 1 and P is isomorphic to a subgroup of AutT ;

(ii) m ≥ 2 and P is a group of diagonal type with n = |T |;(iii) m ≥ 2 and for some proper divisor d of m and some primitive group T with a so-

cle isomorphic to T d , P is isomorphic to a subgroup of the wreath product T wr S Ω, |Ω| =

m/d with the product action, and n = lm/d , where l is the degree of T ;

(iv) m≥

6 , H is regular and n =

|T

|m.

A complete proof of the O’Nan-Scott theorem can be found in the reference [DiM1].

It should be noted that the O’Nan-Scott theorem is a useful result for research problems

related with permutation groups. By Corollary 2.5.3, a finite primitive group P has a

socle H T m, a direct product of m copies of some simple group T . Applying this result

enables one to divide a problem into the following five types in general:



Sec.2.6 Local Action and Extension Groups 69

1. Affine Type: H is an elementary Abelian p-group, n = pm and P is a subgroup of

AGLm( p) containing the translations.

2. Regular Non-Abelian Type: H and T are non-Abelian, n = |T |m, m ≥ 6 and the group

P can be constructed as a twisted wreath product.

3. Almost Simple Type: H is simple and P ≤ Aut H .

4. Diagonal Type: H = T m with m ≥ 2, n = |T |m−1 and P is a subgroup of a wreath

product with the diagonal action.

5. Product Type: H = T m with m = rs, s > 1. There is a primitive non-regular group T

with socle T r and of type in Cases 3 or 4 such that P is isomorphic to a subgroup of the

wreath product T wr S ∆, |∆| = s with the product action.

All these types are contributed to applications of O’Nan-Scott theorem, particularlyfor the classification of symmetric graphs in Chapter 3.

§2.6 LOCAL ACTION AND EXTENDED GROUPS

Let ( G ; O ) be a multigroup with G =m

i=1

G i, O = i|1 ≤ i ≤ m and Ω =m

i=1Ωi a set. An

action (ϕ, ι) of (

G ;

O ) on

Ω is defined to be a homomorphism

(ϕ, ι) : ( G ; O ) →m

i=1

S Ωi

such that ϕ|Ωi: G i → S Ωi

is a homomorphism, i.e., for ∀ x ∈ Ωi, ϕ(h) : x → xh with

conditions following hold,

xhig = xhι(i) xg, h, g ∈ H i

for any integer 1 ≤ i ≤ m. We say ϕ|Ωithe local action of (ϕ, ι) on

Ω for integers 1 ≤ i ≤ m.

2.6.1 Local Action Group. If the multigroup ( G ; O ) is in fact a permutation group Pwith Ω =

mi=1

Ωi, we call such a P to be a local action group on Ωi for integers 1 ≤ i ≤ m.

In this case, a local action of P on Ω is determined by

ΩPi = Ωi and (Ω \ Ωi)

P = Ω \ Ωi

for integers 1 ≤ i ≤ m.




If the local action of P on Ωi is transitive or regular, then we say it is a locally

transitive group or locally regular group on Ωi for an integer 1 ≤ i ≤ m. We know the

following necessary condition for locally transitive or regular groups by Theorem 2.2.1

and Corollary 2.2.1.

Theorem 2.6.1 Let P be a group action on Ω =m

i=1Ωi and H ≤ P. Then H is locally

transitive only if there is an integer k 0, 1 ≤ k 0 ≤ m such that |Ωk 0 | | |H |. Furthermore, if

it is locally regular, then there is an integer l0, 1 ≤ l0 ≤ m such that |Ωi0| = |H |.

Let P be a group locally acting on Ω, where Ω =m

i=1Ωi. If there are integers

k , i, k ≥ 2, 1 ≤ i ≤ m such that the action of P on Ωi is k -transitive or sharply k -transitive,

we say it is a locally k-transitive group or locally sharply k-transitive group on

Ω. The

following necessary condition for locally k -transitive or sharply groups is by Theorems2.2.3– 2.2.5.

Theorem 2.6.2 Let P be a group action on Ω =m

i=1Ωi and H ≤ P. Then H is locally

k-transitive only if there is an integer i0, 1 ≤ i0 ≤ m such that for ∀a ∈ Ωi0 , H a is

(k − 1)-transitive acting on Ω \ a. Particularly, |Ωi0|(|Ωi0

| − 1) · · · (|Ωi0| − k + 1) | |H |.

Furthermore, if it is locally sharply k-transitive, then there is an integer j 0, 1 ≤ j0 ≤ m

such that |Ω j0|(|Ω j0

| − 1) · · · (|Ω j0| − k + 1) = |H |.

Theorems 2.6.1 and 2.6.2 enables us to know what kind subgroups maybe locally

action groups.

Example 2.6.1 Let P be a permutation group with

P = 1P , (1, 2, 3, 4, 5), (1, 4, 2, 5, 3), (1, 5, 4, 3, 2)

(2, 3, 5, 4), (1, 3, 2, 5), (1, 5, 4, 3), (1, 2, 4, 3), (1, 4, 5, 2)

(2, 4, 5, 3), (1, 4, 3, 5), (1, 2, 5, 4), (1, 5, 2, 3), (1, 3, 4, 2)

(2, 5)(3, 4), (1, 5)(2, 4), (1, 4)(2, 3), (1, 3)(4, 5), (1, 2)(3, 5)

ThenH = 1P , (1, 2, 3, 4, 5), (1, 4, 2, 5, 3), (1, 5, 4, 3, 2),

T = 1P , (1, 2, 3, 4), (1, 3)(2, 4), (1, 4, 3, 2)both are subgroups of P. Notice that |H | = 5, |T | = 4. We know that H and T are

transitive acting on Ω = 1, 2, 3, 4, 5 and ∆ = 1, 2, 3, 4, respectively. But none of them

is k -transitive for k ≥ 2.




Corollary 2.6.1 Let P be a group action on Ω =m

i=1Ωi , H ≤ P . For integers i, 1 ≤ i ≤

m and k ≥ 1 , if |Ωi|(|Ωi| − 1)(|Ωi| − 2) · · · (|Ωi| − k + 1) is not a divisor of |H | , then (H ; )

is not locally k-transitive on Ωi.

For a local action group P on Ω with Ω =m

i=1Ωi, if there is an integer i, 1 ≤ i ≤ m

such that the action of P on Ωi is primitive, we say it is a locally primitive group on Ω.

The following condition for locally primitive group is by Theorems 2.5.4.

Theorem 2.6.3 Let P be a local action group on Ω =m

i=1Ωi with H < P . Then (H ; )

is locally primitive if and only if there is an integer l, 1 ≤ l ≤ m such that H action on

Ωl is transitive and H a is maximal for ∀a ∈ Ωl.

2.6.2 Action Extended Group. Conversely, letP

be a permutation group action on Ω,

∆ a set with ∆ ∩ Ω = ∅. A permutation group P action on Ω ∪ ∆ is an action extended

of P on Ω if ( P)∆ = P, and k-transitive extended or primitive extended if P action on

Ω∪ ∆ is k -transitive for an integer k ≥ 1 or primitive. Particularly, if |∆| = 1, such a action

extended group is called one-point extended on P .

The following result is simple.

Theorem 2.6.4 Let P be a permutation group action on Ω , ∆ ∩ Ω = ∅ , k ≥ 1 an integer

and

P an extension of P action on ∆ ∪ Ω. If

(1) P is k-transitive on ∆;

(2) there are k elements x1, x2, · · · , xk ∈ ∆ such that for l elements y1, y2, · · · , yl ∈ Ω ,

where 1 ≤ l ≤ k there exists an element πl ∈ P with

yπl

i= xi f or 1 ≤ i ≤ l b u t xπ

i = xi i f l + 1 ≤ i ≤ k ,

hold, then P is k-transitive extended on ∆ ∪ Ω.

Proof Let xi, yi, 1 ≤ i ≤ k be 2k elements in Ω ∪ ∆. Firstly, we prove that for any

choice of x1, x2, · · · , xk ∈ Ω∪ ∆, there always exists an element θ ∈ P such that all xθ i ∈ ∆

for 1 ≤ i ≤ k . If x1, x2, · · · , xk ∈ ∆, there are no words need to say. Not loss of generality,

we assume that x1, x2, · · · , xs ∈ Ω but xs+1, xs+2, · · · , xk ∈ ∆ for an integer 1 ≤ s ≤ k . Then

by the assumption (2), there is an element πs ∈ P such that xπs

i∈ ∆ for 1 ≤ i ≤ s but

xπs

i= xi for s + 1 ≤ i ≤ k . Whence, x

πs

i∈ ∆ for 1 ≤ i ≤ k , i.e., θ = πs is for our objective.

Similarly, there also exists an element τ ∈ P such that yτi∈ ∆ for 1 ≤ i ≤ k .




Applying the assumption (1), there is an element π ∈ P such that ( xθ i )π = yτ

i for

integers 1 ≤ i ≤ k . Consequently, we know that

xθπτ−1

i= y

i for 1 ≤i≤

k .


Particularly, if k = 1, we get the following conclusion for transitive extended by

Theorem 2.6.4.

Corollary 2.6.2 Let P be a permutation group action on Ω , ∆ ∩ Ω = ∅ and P an

extension of P action on ∆ ∪ Ω. If

(1) P is transitive on ∆;

(2) there is one element x ∈ ∆ such that for any element y ∈ Ω , there exists an

element π ∈ P with yπ = x hold,

then P is transitive extended on ∆ ∪ Ω.

Furthermore, if P is one-point extended of P, we get the following result.

Corollary 2.6.3 Let P be an one-point extension of P action on Ω by x Ω. For

∀ y ∈ Ω , if there exists an element π ∈

P such that yπ = x, then

P is transitive extended

of P .

These conditions in Corollaries 2.6.2–2.6.3 is too strong. In fact, we improve condi-

tions in them as in the following result.

Theorem 2.6.5 Let P be a permutation group action on Ω with orbits B1, B2, · · · , Bm ,

∆ ∩ Ω = ∅ and P = P; Q ,

with Q = ( x, yi), 1 ≤ i ≤ m; ( x′, z), x′ ∈ ∆, x′ x , where x ∈ ∆, yi ∈ Bi , z = x or yi

for 1

≤i

≤m. Then P is transitive extended. Furthermore, if P is transitive on Ω or

∆ = x , i.e., P is one-point extension of P , then

P = P ; ( x, y), ( x′, z), x′ ∈ ∆, x′ x or P; ( x, yi), 1 ≤ i ≤ m

with y ∈ Ω , z = x or y is transitive extended of P on Ω ∪ ∆ or Ω ∪ x.

Proof We only prove the first assertion since all others are then followed.




Firstly, for ∀ zi ∈ Bi, z j ∈ B j, let zσ1

i= yi and zσ2

j= y j, σi, σ j ∈ P . Then

zσi( x, yi)( x, y j )σ j

i= z j. Now if x1, x2 ∈ ∆, by definition x

( x1 , x)( x2 , x)1 = x2, or x

( x1 , x)( x, yi)( yi , x2)

1 = x2,

or x( x1 , yi)( x2 , yi)

1 = x2, or x( x1 , yi)( yi , x)( x, y j )( y j , x2)

1 = x2 if ( x1, x), ( x2, x), or ( x1, x), ( x, yi), ( yi, x2), or

( x1, yi), ( x2, yi), or ( x1, yi), ( yi, x), ( x, y j), ( y j, x2) ∈ P. Finally, if xi ∈ ∆ and z j ∈ B j, let xσ

i= x and z

ς

j= y j. Then x

σ( x, y j )ς

i= z j.

Therefore, P is transitive extended on Ω ∪ ∆.

The k-transitive number trank

(P ; ∆) of a permutation group P action on Ω by a

set ∆ with ∆ ∩ Ω = ∅ is defined to be the minimum number of involutions appeared in

permutations presented by product of inventions added to P such that P is k -transitive

extended of P on Ω∪∆. Particularly, if k = 1, we abbreviate trank

(P ; ∆) to tran(P ; ∆).

We know the number (P; ∆) in the following result.

Theorem 2.6.6 Let P be a permutation group action on Ω with an orbital set Orb(Ω) ,

∆ ∩ Ω = ∅ and P an extended action of P on ∆ ∪ Ω. Then

tran(P; ∆) = |∆| + |Orb(Ω)| − 1.

Furthermore, if P is transitive or P is one-point extension of P , then

tran(P; ∆) = |∆| or |Orb(Ω)|.

Proof Let x ∈ ∆ ∪ Ω be a chosen element. denoted by A[ x] all elements determined

by

A[ x] = y| xπ = y, ∀π ∈ P.

If P is a transitive extended action of P on ∆ ∪ Ω, there must be A[ x] = ∆ ∪ Ω. Enumer-

ating all inventions appeared in permutations π presented by product of inventions such

that xπ = y ∈ A[ x], we know that

tran(P; ∆) ≥ |∆| + |Orb(Ω)| − 1.

Applying Theorem 2.6.5, we get that

tran(P; ∆) ≤ |∆| + |Orb(Ω)| − 1.

Whence,

tran(P; ∆) = |∆| + |Orb(Ω)| − 1.




Notice that |Orb(Ω)| = 1 or |∆| = 1 if P is transitive or P is one-point extension of P.

We therefore find that

tran(P; ∆) = |∆| or |Orb(Ω)|

if P is transitive or P is one-point extended.

Now we turn our attention to primitive extended groups. Applying Theorem 2.5.3,

we have the following result.

Theorem 2.6.7 Let P be a permutation group action on Ω and ∆ a nonempty set with

∆ ∩ Ω = ∅. Then there exist primitive extended permutation groups P of P action on

Ω ∪ ∆ if |∆| ≥ 2 or |∆| = 1 but P is transitive on Ω.

Proof Let B1, B2, · · · , Bm be orbits of P action on Ω. Define

P = P ; ( x, yi), 1 ≤ i ≤ m; ( x′, x), x′ ∈ ∆, x′ x ,

where x ∈ ∆, yi ∈ Bi. Then P is 2-transitive extended of P by Theorem 2.6.4 if |∆| ≥ 2.

Notice that P x = P. If ∆ = x and P is transitive on Ω, we also know that P is

2-transitive extended of P by Theorem 2.2.3. Whence, we know that P is primitive

extended of P on Ω ∪ ∆ by Theorem 2.5.3 in each case.

2.6.3 Action MultiGroup. Let

P be a permutation multigroup action on

Ω with

P =

m

i=1

Pi, Ω =m

i=1

Ωi and for each integer i, 1

≤i

≤m, the permutation group Pi acts on Ωi.

Such a permutation multigroup P is said to be globally k-transitive for an integer k ≥ 1

if for any two k -tuples x1, x2, · · · , xk ∈ Ωi and y1, y2, · · · , yk ∈ Ω j, where 1 ≤ i, j ≤ m, there

are permutations π1, π2, · · · , πn such that

xπ1π2···πn

1 = y1, xπ1π2···πn

2 = yi, · · · , xπ1 π2···πn

k = yk .

For simplicity, we abbreviate the globally 1-transitive to that globally transitive of a per-

mutation multigroup.

Remark 2.6.1: There are no meaning if we define the globally k -transitive on two k -tuples x1, x2, · · · , xk ∈ Ω, y1, y2, · · · , yk ∈ Ω in a permutation multigroup P because there

are no definition for the actions xπl

if xl Ωi but π ∈ Pi, 1 ≤ i ≤ m, where 1 ≤ l ≤ k .

Theorem 2.6.8 Let P be a permutation multigroup action on Ω with P =m

i=1

Pi, Ω =

mi=1

Ωi , where each permutation group Pi transitively acts on Ωi for each integers 1 ≤ i ≤




m. Then P is globally transitive on Ω if and only if for any integer i, 1 ≤ i ≤ m, there

exists an integer j, 1 ≤ j ≤ m, j i such that

ΩiΩ j

∅.

Proof If P is globally transitive action on Ω, by definition for x ∈ Ωi and y Ωi,

1 ≤ i ≤ m, there are elements π1, π2, · · · , πn ∈ P such that

xπ1π2···πn = y.

Not loss of generality, we assume π1, π2, · · · , πl−1 ∈ Pi but πl, πl+1, · · · , πn Pi, i.e., l be

the least integer such that πl Pi. Let πl ∈ P j. Notice that Pi, P j act on Ωi and Ω j,

respectively. We get that x

π1π2

···πi

∈Ω

i ∩Ω

j, i.e.,

Ωi

Ω j ∅.

Conversely, if for any integer i, 1 ≤ i ≤ m, there always exists an integer j, 1 ≤ j ≤m, j i such that

Ωi

Ω j ∅,

let x ∈ Ωi and y Ωi. Then there exist integers l1, l2, · · · , ls such that

ΩiΩl1 ∅, Ωl1 Ωl2 ∅, · · · , Ωls−1 Ωls ∅.

Let x, x1 ∈ Ωi

Ωl1

, x2 ∈ Ωl1

Ωl2

, · · ·, xs ∈ Ωls−1

Ωls

, y ∈ Ωlsand π1 ∈ P1, π2 ∈ Pl1

,

· · ·, πs−1 ∈ Pls−1, πs ∈ Pls

such that xπ1 = xl1, xπ2

l1= xl2

,· · ·, xπs−1

ls−1= xls

, xπs

ls= y by the

transitivity of Pi, 1 ≤ i ≤ m. Therefore, we find that

xπ1π2···πs = y.


The condition of transitivity on each permutation Pi, 1 ≤ i ≤ m in Theorem 2.6.8 is

not necessary for the globally transitive of P on Ω, such as those shown in the following

example.

Example 2.6.2 Let P be a permutation multigroup action on Ω with

P = P1

P2 and Ω = 1, 2, 3, 4, 5, 6, 7, 8

1, 2, 5, 6, 9, 10, 11, 12,




where P1 = (1, 2, 3, 4), (5, 6, 7, 8) and P2 = (1, 5, 9, 10), (2, 6, 11, 12), i.e.,

P1 = 1P1, (13)(24), (1, 2, 3, 4), (1, 4, 3, 2),

(5, 7)(6, 8), (5, 8, 7, 6), (5, 6, 7, 8),

(13)(24)(5, 7)(6, 8), (13)(24)(5, 6, 7, 8), (13)(24)(5, 8, 7, 6)

(1, 2, 3, 4)(5, 7)(6, 8), (1, 2, 3, 4)(5, 6, 7, 8), (1, 2, 3, 4)(5, 8, 7, 6)

(1, 4, 3, 2)(5, 7)(6, 8), (1, 4, 3, 2)(5, 6, 7, 8), (1, 4, 3, 2)(5, 8, 7, 6)

and

P2 = 1P2, (1, 9)(5, 10), (1, 5, 9, 10), (1, 10, 9, 5)

(2, 11)(6, 12), (2, 6, 11, 12), (2, 12, 11, 6)

(1, 9)(5, 10)(2, 11)(6, 12), (1, 9)(5, 10)(2, 6, 11, 12), (1, 9)(5, 10)(2, 12, 11, 6)

(1, 5, 9, 10)(2, 11)(6, 12), (1, 5, 9, 10)(2, 6, 11, 12), (1, 5, 9, 10)(2, 12, 11, 6)

(1, 10, 9, 5)(2, 11)(6, 12), (1, 10, 9, 5)(2, 6, 11, 12), (1, 10, 9, 5)(2, 12, 11, 6).

Calculation shows that P is transitive on Ω, i.e., for any element, for example 1 ∈ Ω,

1P = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Generally, we know the following result on the globally transitive of permutation

multigroup, a generalization of Theorem 2.6.8 motivated by Example 2.6.2.

Theorem 2.6.9 Let P be a permutation multigroup action on Ω with P =m

i=1

Pi, Ω =

mi=1

Ωi , where each permutation group Pi acts on Ωi with orbits Bi j, 1 ≤ j ≤ |Orb(Ωi)| for

integers 1 ≤ i ≤ m. Then P is globally transitive on Ω if and only if for integer i, j, 1 ≤i ≤ m, 1 ≤ j ≤ |Orb(Ωi)| , there exist integers k , 1 ≤ k ≤ m, 1 ≤ l ≤ |Orb(Ωk )|, k i such

that

Ωi j

Ωkl ∅.

Proof Define a multiset

Ω =

mi=1

Ωi =

mi=1

|Orb(Ωi)| j=1

Bi j

.

Then Pi acts on each Bi j is transitive by definition for 1 ≤ i ≤ m, 1 ≤ j ≤ |Orb(Ωi)| and

the result is followed by Theorem 2.6.8.



Sec.2.7 Remarks 77

Counting elements in each Ωi, 1 ≤ i ≤ m, we immediately get the following conse-

quence by Theorem 2.6.9.

Corollary 2.6.3 Let P be a permutation multigroup globally transitive action on Ωwith P =m

i=1

Pi, Ω =m

i=1

Ωi , where each permutation group Pi acts on Ωi with orbits

Bi j, 1 ≤ j ≤ |Orb(Ωi)| for integers 1 ≤ i ≤ m. Then for any integer i, 1 ≤ i ≤ m,

|Ω \ Ωi| ≥ |Orb(Ωi)|, particularly, if m = 2 then

|Ω1| ≥ |Orb(Ω2)| and |Ω2| ≥ |Orb(Ω1)|.A permutation multigroup

P =

m

i=1

Pi action on

Ω =

m

i=1

is said to be globally

primitive if there are no proper subsets A ⊂ Ω, | A| ≥ 2 such that either A = A

π

or A ∩ Aπ = ∅ for ∀π ∈ P provided aπ existing for ∀a ∈ A.

Theorem 2.6.10 A permutation multigroup P =m

i=1

Pi action on Ω =m

i=1

is globally

primitive if and only if Pi action on Ωi is primitive for any integer 1 ≤ i ≤ m.

Proof If P action on Ω is globally primitive, by definition we know that there are

no proper subsets A ⊂ Ωi, | A| ≥ 2 such that either A = Aπ or A ∩ Aπ = ∅ for ∀π ∈ Pi,

where 1 ≤ i ≤ m. Whence, each Pi primitively acts on Ωi.

Conversely, if each Pi action on Ωi is primitive for integers 1

≤i

≤m, then there

are no proper subsets A ⊂ Ωi, | A| ≥ 2 such that either A = Aπ or A ∩ Aπ = ∅ for ∀π ∈ Pi

for 1 ≤ i ≤ m by definition. Now let π ∈ Pi for an integer i, 1 ≤ i ≤ m. Notice that Aπ

is existing for ∀ A ⊂ Ω if and only if A ⊂ Ωi. Consequently, P action on Ω is globally

primitive by definition.

Combining Theorems 2.6.10 with 2.5.4, we get the following consequence.

Corollary 2.6.4 Let P =m

i=1Pi be a permutation multigroup action on Ω =

mi=1

, where

Pi is transitive and (Pi)a is maximal for ∀a ∈ Ωi , 1 ≤ i ≤ m. Then P is globally

primitive action on Ω.

§2.7 REMARKS

2.7.1 There are many monographs on action groups such as those of [Wie1] and [DiM1].

In fact, every book on group theory partially discusses action groups with applications.




These materials in Sections 2.1, 2.2 2.3 and 2.5 are mainly extracted from [Wan1], [Rob1]

and [DiM1], particularly, the O’Nan-Scott theorem on primitive groups.

2.7.2 A central but difficult problem in group theory is to classify groups of order n for

any integer n ≥ 1. The Sylow’s theorem on p-groups enables one to see a glimmer on

classifying p-groups. However, this problem is also difficult in general. Today, we can

only find the classification of p-groups with small power (See [Xum1] and [Zha1] for

details). In fact, these techniques used for classifying p-groups are nothing but the group

actions, i.e., application of action groups.

2.7.3 These permutation multigroups in Section 2.6 is in fact action multigroups, a kind

of Smarandache multi-spaces first discussed in [Mao21] and [Mao25]. These concep-

tions such as those of locally k -transitive, locally primitive, k -transitive extended, prim-itive extended, globally transitive and globally primitive are first presented in this book.

Certainly, there are many open problems on permutation multigroups, for example, for a

permutation group P action on Ω , is there always an extended primitive action of P on

Ω ∪ ∆ for a set ∆, ∆ ∩ Ω = ∅? Can we characterize such permutation groups P or such

sets ∆?

2.7.4 Theorems 2.6.8 and 2.6.9 completely determine the globally transitive multigroups.

However, we can also find a more simple characterization by graphs in Chapter 3, in where

we clarify the property of globally transitive is nothing but the connectedness on graphs.

In fact, these conditions in Theorems 2.6.8 and 2.6.9 are essentially enables one to find a

spanning tree, a kind of most simple connected graph on Ω.



CHAPTER 3.

Graph Groups

An immediate applying field of action groups is to that of graphs for themeasily to handle by intuition. By definition, a graph group is a subgroup of

the automorphism group of a graph viewed as a permutation group of its ver-

tices. In fact, graphs has a nice mathematical structure on objectives. Usu-

ally, the investigation on such structures enables one to find new important

results in mathematics. For example, the well-known Higman-Sims group,

one of these 26 sporadic simple groups was found by that of graph groups

in 1968. Topics covered in the first 4 sections including graphs with opera-

tions, graph properties with results, Smarandachely graph properties, graph

groups, vertex-transitive graphs, edge-transitive graphs, arc-transitive graphs,

semi-arc groups with semi-arc transitive graph, · · ·, etc.. A graph is itself

a Smarandache multi-space by definition, which naturally provide us a nice

source for get multigroups. In Section 3.5, we show how to get mutligroups

on graphs, also find new graph invariants by that of graph multigroups, which

will be useful for research graphs and getting localized symmetric graphs.



80 Chap.3 Graph Groups

§3.1 GRAPHS

3.1.1 Graph. A graph G is an ordered 3-tuple (V , E ; I ), where V , E are finite sets, V ∅

and I : E → V × V . Call V the vertex set and E the edge set of G, denoted by V (G)and E (G), respectively. An elements v ∈ V (G) is incident with an element e ∈ E (G)

if I (e) = (v, x) or ( x, v) for an x ∈ V (G). Usually, if (u, v) = (v, u), denoted by uv or

vu ∈ E (G) for ∀(u, v) ∈ E (G), then G is called to be a graph without orientation and

abbreviated to graph for simplicity. Otherwise, it is called to be a directed graph with an

orientation u → v on each edge (u, v).

The cardinal numbers of |V (G)| and | E (G)| are called its order and size of a graph G,

denoted by |G| and ε(G), respectively.

Let G be a graph. We can represent a graph G by locating each vertex u in G by apoint p(u), p(u) p(v) if u v and an edge (u, v) by a curve connecting points p(u) and

p(v) on a plane R2, where p : G → P is a mapping from the V (G) to R2.

For example, a graph G = (V , E ; I ) with V = v1, v2, v3, v4, E = e1, e2, e3, e4, e5,

e6, e7, e8, e9, e10 and I (ei) = (vi, vi), 1 ≤ i ≤ 4; I (e5) = (v1, v2) = (v2, v1), I (e8) = (v3, v4) =

(v4, v3), I (e6) = I (e7) = (v2, v3) = (v3, v2), I (e8) = I (e9) = (v4, v1) = (v1, v4) can be drawn

on a plane as shown in Fig.3.1.1.

v1 v2

v3v4

e1 e2

e3e4

e5

e6e7

e8

e9 e10

Fig. 3.1.1

Let G = (V , E ; I ) be a graph. For ∀e ∈ E , if I (e) = (u, u), u ∈ V , then e is called a loop,

For example, edges e1 − e4 in Fig.3.1.1. For non-loop edges e1, e2 ∈ E , if I (e1) = I (e2),

then e1, e2 are called multiple edges of G. In Fig.3.1.1, edges e6, e7 and e9, e10 are multiple

edges. A graph is simple if it is loopless without multiple edges, i.e., I (e) = (u, v) implies

that u v, and I (e1) I (e2) if e1 e2 for ∀e1, e2 ∈ E (G). In the case of simple graphs, an

edge (u, v) is commonly abbreviated to uv.



Sec.3.1 Graphs 81

A walk of a graph G is an alternating sequence of vertices and edges u1, e1, u2, e2,

· · · , en, un with ei = (ui, ui+1) for 1 ≤ i ≤ n. The number n is called the length of the

walk . A walk is closed if u1 = un+1, and opened , otherwise. For example, the sequence

v1e1v1e5v2e6v3e3v3e7v2e2v2 is a walk in Fig.1.3.1. A walk is a trail if all its edges aredistinct and a path if all the vertices are distinct also. A closed path is usually called a

circuit or cycle. For example, v1v2v3v4 and v1v2v3v4v1 are respective path and circuit in

Fig.3.1.1.

A graph G = (V , E ; I ) is connected if there is a path connecting any two vertices in

this graph. In a graph, a maximal connected subgraph is called its a component .

Let G be a graph. For ∀u ∈ V (G), the neighborhood N G(u) of the vertex u in G is

defined by N G(u) = v|∀(u, v) ∈ E (G). The cardinal number | N G(u)| is called the valency

of vertex u in G and denoted by ρG(u). A vertex v with ρG(v) = 0 is an isolated vertexand ρG(v) = 1 a pendent vertex. Now we arrange all vertices valency of G as a sequence

ρG(u), ρG(v), · · · , ρG(w) with ρG(u) ≥ ρG(v) ≥ · · · ≥ ρG(w), and denote ∆(G) = ρG(u),

δ(G) = ρG(w) and call then the maximum or minimum valency of G, respectively. This

sequence ρG(u), ρG(v), · · · , ρG(w) is usually called the valency sequence of G. If ∆(G) =

δ(G) = r , such a graph G is called a r-regular graph. For example, the valency sequence

of graph in Fig.3.1.1 is (5, 5, 5, 5), which is a 5-regular graph.

By enumerating edges in E (G), the following equality is obvious.

u∈V (G)

ρG(u) = 2| E (G)|.

A graph G with a vertex set V (G) = v1, v2, · · · , v p and an edge set E (G) = e1, e2, · · · ,

eq can be also described by those of matrixes. One such matrix is a p × q adjacency ma-

trix A(G) = [ai j] p×q, where ai j = | I −1(vi, v j)|. Thus, the adjacency matrix of a graph G is

symmetric and is a 0, 1-matrix having 0 entries on its main diagonal if G is simple. For

example, the matrix A(G) of the graph in Fig.3.1.1 is

A(G) =

1 1 0 2

1 1 2 0

0 2 1 1

2 0 1 1

Let G1 = (V 1, E 1; I 1) and G2 = (V 2, E 2; I 2) be two graphs. They are identical, denoted

by G1 = G2 if V 1 = V 2, E 1 = E 2 and I 1 = I 2. If there exists a 1 − 1 mapping φ : E 1 →




E 2 and φ : V 1 → V 2 such that φ I 1(e) = I 2φ(e) for ∀e ∈ E 1 with the convention that

φ(u, v) = (φ(u), φ(v)), then we say that G1 is isomorphic to G2, denoted by G1 G2 and

φ an isomorphism between G1 and G2. For simple graphs H 1, H 2, this definition can be

simplified by (u, v) ∈ I 1( E 1) if and only if (φ(u), φ(v)) ∈ I 2( E 2) for ∀u, v ∈ V 1.For example, let G1 = (V 1, E 1; I 1) and G2 = (V 2, E 2; I 2) be two graphs with

V 1 = v1, v2, v3, E 1 = e1, e2, e3, e4,

I 1(e1) = (v1, v2), I 1(e2) = (v2, v3), I 1(e3) = (v3, v1), I 1(e4) = (v1, v1)

and

V 2 = u1, u2, u3, E 2 = f 1, f 2, f 3, f 4,

I 2( f 1) = (u1, u2), I 2( f 2) = (u2, u3), I 2( f 3) = (u3, u1), I 2( f 4) = (u2, u2),

i.e., those graphs shown in Fig.3.1.2.

u1

v2v3

e1

e2

e3

e4

G1

v1

u2u3

f 1 f 2

f 3

f 4

G2

Fig. 3.1.2

Define a mapping φ : E 1

V 1 → E 2

V 2 by φ(e1) = f 2, φ(e2) = f 3, φ(e3) =

f 1, φ(e4) = f 4 and φ(vi) = ui for 1 ≤ i ≤ 3. It can be verified immediately that

φ I 1(e) = I 2φ(e) for ∀e ∈ E 1. Therefore, φ is an isomorphism between G1 and G2, i.e.,

G1 and G2 are isomorphic.

A graph H = (V 1, E 1; I 1) is a subgraph of a graph G = (V , E ; I ) if V 1 ⊆ V , E 1 ⊆ E

and I 1 : E 1 → V 1 × V 1. We use H ≺ G to denote that H is a subgraph of G. For example,

graphs G1, G2, G3 are subgraphs of the graph G in Fig.3.1.3.

u1 u2

u3u4

G

u1 u2

u3 u4

u1 u2

u3 u4

G1 G2 G3

Fig. 3.1.3



Sec.3.1 Graphs 83

For a nonempty subset U of the vertex set V (G) of a graph G, the subgraph U of G

induced by U is a graph having vertex set U and whose edge set consists of these edges

of G incident with elements of U . A subgraph H of G is called vertex-induced if H U

for some subset U of V (G). Similarly, for a nonempty subset F of E (G), the subgraph F induced by F in G is a graph having edge set F and whose vertex set consists of vertices

of G incident with at least one edge of F . A subgraph H of G is edge-induced if H F for some subset F of E (G). In Fig.3.1.3, subgraphs G1 and G2 are both vertex-induced

subgraphs u1, u4, u2, u3 and edge-induced subgraphs (u1, u4), (u2, u3). For

a subgraph H of G, if |V ( H )| = |V (G)|, then H is called a spanning subgraph of G. In

Fig.3.1.3, the subgraph G3 is a spanning subgraph of the graph G.

K (4, 4) K 6

Fig.3.1.4

A graph G is n-partite for an integer n ≥ 1, if it is possible to partition V (G) into n

subsets V 1, V 2,

· · ·, V n such that every edge joints a vertex of V ito a vertex of V j, j i, 1

≤i, j ≤ n. A complete n-partite graph G is such an n-partite graph with edges uv ∈ E (G) for

∀u ∈ V i and v ∈ V j for 1 ≤ i, j ≤ n, denoted by K ( p1, p2, · · · , pn) if |V i| = pi for integers

1 ≤ i ≤ n. Particularly, if |V i| = 1 for integers 1 ≤ i ≤ n, such a complete n-partite graph

is called complete graph and denoted by K n. In Fig.3.1.4, we can find the bipartite graph

K (4, 4) and the complete graph K 6. Usually, a complete subgraph of a graph is called a

clique, and its a k -regular vertex-spanning subgraph also called a k-factor .

3.1.2 Graph Operation. A union G1

G2 of graphs G1 with G2 is defined by

V (G1G2) = V 1V 2, E (G1

G2) = E 1 E 2, I ( E 1 E 2) = I 1( E 1) I 2( E 2).

A graph consists of k disjoint copies of a graph H , k ≥ 1 is denoted by G = kH . As an

example, we find that

K 6 =

5i=1

S 1.i




for graphs shown in Fig.3.1.5 following

1

2 3

4 5

62

34

5

63

4

5

64

5

65

6

S 1.5 S 1.4 S 1.3 S 1.2 S 1.1

Fig. 3.1.5

and generally, K n =n−1i=1

S 1.i. Notice that kG is a multigraph with edge multiple k for any

integer k , k ≥ 2 and a simple graph G.

A complement G of a graph G is a graph with vertex set V (G) such that vertices are

adjacent in G if and only if these are not adjacent in G. A join G1 + G2 of G1 with G2 is

defined by

V (G1 + G2) = V (G1)

V (G2),

E (G1 + G2) = E (G1)

E (G2)(u, v)|u ∈ V (G1), v ∈ V (G2)

and

I (G1 + G2) = I (G1)

I (G2)

I (u, v) = (u, v)|u ∈ V (G1), v ∈ V (G2).

Applying the join operation, we know that K (m, n) K m + K n. A Cartesian product G1 ×G2 of graphs G1 with G2 is defined by V (G1 × G2) = V (G1) × V (G2) and two vertices

(u1, u2) and (v1, v2) of G1×G2 are adjacent if and only if either u1 = v1 and (u2, v2) ∈ E (G2)

or u2 = v2 and (u1, v1) ∈ E (G1). For example, K 2 × P6 is shown in Fig.3.1.6 following.

u

v

1 2 3 4 5K 2

6

P6

K 2 × P6

u1 u2 u3 u4 u5 u6

v1 v2 v3 v4 v5 v6

Fig.3.1.6



Sec.3.1 Graphs 85

3.1.3 Graph Property. A graph property P is in fact a graph family

P = G1, G2, G3, · · · , Gn, · · ·

closed under isomorphism, i.e., Gϕ ∈ P for any isomorphism on a graph G ∈ P. We

alphabetically list some graph properties and results without proofs following.

Colorable. A coloring of a graph G by colors in C is a mapping ϕ : C →V (G) ∪ E (G) such that ϕ(u) ϕ(v) if u is adjacent or incident with v in G. Usually, a

coloring ϕ|V (G) : C → V (G) is called a vertex coloring and ϕ| E (G) : C → E (G) an edge

coloring. A graph G is n-colorable if there exists a color set C for an integer n ≥ |C |. The

minimum number n for which a graph G is vertex n-colorable, edge n-colorable is called

the vertex chromatic number or edge chromatic number and denoted by χ(G) or χ1(G),

respectively. The following result is well-known for colorable of a graph.

Theorem 3.1.1 Let G be a connected graph. Then

(1) χ(G) ≤ ∆(+) + 1 and with the equality hold if and only if G is either an odd

circuit or a complete graph; (Brooks theorem)

(2) χ1(G) = ∆(G) or ∆(G) + 1; (Vizing theorem)

Theorem 3.1.1(2) enables one to classify graphs into Class 1, Class 2 by χ1(G) =

∆(G) or χ1(G) = ∆(G) + 1, respectively.

Connectivity. For an integer k ≥ 1, a graph G is said to be k-connected if removing

elements in X ⊂ V (G)∪ E (G) with | X | = k still remains a connected graph G − X . Usually,

we call G to be vertex k-connected or edge k-connected if X ⊂ V (G) or X ⊂ E (G) and

abbreviate vertex k -connected to k-connected in reference. The minimum cardinal number

of X ⊂ V (G) or X ⊂ E (G) is defined to be the connectivity or edge-connectivity of G,

denoted respective by κ (G), κ 1(G). A fundamental result for characterizing connectivity

of a graph is the Menger theorem following.

Theorem 3.1.2(Menger) Let u and v be non-adjacent vertices in a graph G. Then the

minimum number of vertices that separate u and v is equal to that the maximum number

of internally disjoint u − v paths in G.

Then we can characterize k -connected or k -edge-connected graphs following.

Theorem 3.1.3 Let G be a non-trivial graph. Then




(1) G is k-connected if and only if for ∀u, v ∈ V (G), u v, there are at least k

internally disjoint u − v paths in G. (Whinety)

(2) G is k-edge-connected if and only if for ∀u, v ∈ V (G), u v, there are at least k

edge-disjoint u − v paths in G.

Covering. A subset W ⊂ V (G) ∪ E (G) is independent if any two element in W

is non-adjacent or non-incident. A vertex and an edge in a graph are said to be cover

each other if they are incident and a cover of G is such a subset U ⊂ V (G) ∪ E (G) such

that any element in V (G) ∪ E (G) \ U is incident to an element in U . If U ⊂ V (G) or

U ⊂ E (G), such an independent set is called vertex independent or edge independent and

such a covering a vertex cover or edge cover . Usually, we denote the minimum cardinality

of vertex cover, edge cover of a graph G by α(G) an α1(G) and the maximum cardinality

of vertex independent set, edge independent set by β(G) and β1(G), respectively.

Theorem 3.1.4(Gallai) Let G be a graph of order p without isolated vertices. Then

α(G) + β(G) = p and α1(G) + β1(G) = p.

A dominating set D of a graph G is such a subset D ⊂ V (G) ∪ E (G) such that every

element is adjacent to an element in D. If D ⊂ V (G) or D ⊂ E (G), such a dominating set

D of G is called a vertex or edge dominating set . The minimum cardinality of vertex or

edge dominating set is denoted by σ(G) or σ1(G), called the vertex or edge dominating

number , respectively. The following is obvious by definition.

Theorem 3.1.5 Let G be a graph. Then

σ(G) ≤ α(G) and σ1(G) ≤ β1(G).

Decomposable. A decomposition of a graph G is subgraphs H i; 1 ≤ i ≤ m of G such

that H i = E i for some subset E i ⊂ E (G) with E i ∩ E j = ∅ for j i, 1 ≤ j ≤ m, usually

denoted by

G =

mi=1

H i.

If every H i is a spanning subgraph of G, such a decomposition is called a factorization of

G into factors H i; 1 ≤ i ≤ m. Furthermore, if every H i is k -regular, such a decomposition

is called k-factorable and each H i is a k -factor of G.



Sec.3.1 Graphs 87

u1 u2

u3

u4u5

u6

v1 v2

v3v4

G1 G2

Fig.3.1.7

For example, we know that

G1 = H 1

H 2, and G2 = F 1

F 2

F 3

for graphs G1, G2 in Fig.3.1.8, where H 1 = u1u4, u2u3, u5u6, h2 = u1u6, u2u5, u3u4and F 1 = v1v2, v3v4, F 2 = v1v4, v2v3, F 3 = v1v3, v2v4. Notice that every H i or F i is

1-regular. Such a spanning subgraph in a graph G is called a perfect matching of G.

Theorem 3.1.6(Tutte) A non-trivial graph G has a perfect matching if and only if for

every proper subset S ⊂ V (G) ,

ω(G − S ) ≤ |S |,where ω( H ) denotes the number of odd components in a graph H.

Theorem 3.1.7(Konig) Every k-regular bipartite graph with k ≥ 1 is 1-factorable.

Theorem 3.1.8(Petersen) A non-trivial graph G is 2-factorable if and only if G is 2n-

regular for some integer n ≥ 1.

Embeddable. A graph G is said to be embeddable into a topological space T if there

is a 1 − 1 continuous mapping f : G → T with f ( p) f (q) if p, q V (G). Particularly, if

T is a Euclidean plane R2, we say that G is a planar graph. In a planar graph G, its face

is defined to be that region F in which any simple curve can be continuously deformed in

this region to a single point p ∈ F . For example, the graph in Fig.3.1.8 is a planar graph.

v1 v2

v3v4

u1 u2

u3u4

Fig.3.1.8




whose faces are F 1 = u1u2v3u4u1, F 2 = v1v2v3v4v1, F 3 = u1v1v2u2u1, F 4 = u2v2v3u3u2,

F 5 = u3v3v4u4u3 and F 6 = u4v4v1u1u4. It should be noted that each boundary of a face

in this planar graph is a circuit. Such an embedding graph is called a strong embedded

graph.

Theorem 3.1.9(Euler) Let G be a planar graph with p vertices, q edges and r faces. Then

p − q + r = 2.

An elementary subdivision of a graph G is such a graph obtained from G by removing

some edge e = uv and adding a new vertex and two edges uw, vw. A subdivision of a graph

G is a graph by a succession of elementary subdivision. Define a graph H homeomorphic

from that of G if either H

G or H is isomorphic to a subdivision of G. The followingresult characterizes planar graphs.

Theorem 3.1.10(Kuratowski) A graph is planar if and only if it contains no subgraphs

homeomorphic with K 5 or K (3, 3).

Theorem 3.1.11(The Four Color Theorem) Every planar graph is 4-colorable.

Travelable. A graph G is eulerian if there is a closed trail containing all edges and

is hamiltonian if there is a circuit containing all vertices of G. For example, the graph in

Fig.3.1.6 is with a hamiltonian circuit C =

v1v2v3v4u4u3u2u2v1, but it is not eulerian. Weknow a necessary and sufficient condition for eulerian graphs following.

Theorem 3.1.12(Euler) A graph G is eulerian if and only if ρG(v) ≡ 0(mod2), ∀v ∈ V (G).

But for hamiltonian graphs, we only know some sufficient conditions. For example,

the following results.

Theorem 3.1.13(Chvatal and Erdos) Let G be a graph with at least 3 vertices. If κ (G) ≥ β(G) , then G is hamiltonian.

A closure C (G) of a graph G is the graph obtained by recursively joining pairs of

non-adjacent vertices whose valency sum is at least |G|. Then we know the next result.

Theorem 3.1.14(Bondy and Chatal) A graph is hamiltonian if and only if its closure is

hamiltonian.

Theorem 3.1.15(Tutte) Every 4-connected planar graph is hamiltonian.



Sec.3.1 Graphs 89

3.1.4 Smarandachely Graph Property. A graph property P is Smarandachely if it

behaves in at least two diff erent ways on a graph, i.e., validated and invalided, or only

invalided but in multiple distinct ways. Such a graph with at least one Smarandachely

graph property is called a Smarandachely graph. Here, we only alphabetically list someSmarandachely graph properties and results with some open problems following.

Smarandachely Coloring. Let Λ be a subgraph of a graph G. A Smarandachely

Λ-coloring of a graph G by colors in C is a mapping ϕΛ : C → V (G) ∪ E (G) such

that ϕ(u) ϕ(v) if u and v are elements of a subgraph isomorphic to Λ in G. Similarly,

a Smarandachely Λ-coloring ϕΛ|V (G) : C → V (G) or ϕΛ| E (G) : C → E (G) is called

a vertex Smarandachely Λ-coloring or an edge Smarandachely Λ-coloring. A graph G

is Smarandachely n Λ-colorable if there exists a color set C for an integer n ≥ |C |. The

minimum number n for which a graph G is Smarandachely vertex n Λ-colorable, Smaran-

dachely edge n Λ-colorable is called the vertex Smarandachely chromatic Λ-number or

edge Smarandachely chromatic Λ-number and denoted by χΛ(G) or χΛ1 (G), respectively.

Particularly, if Λ = P2, i.e., an edge, then a vertex Smarandachely Λ-coloring or an edge

Smarandachely Λ-coloring is nothing but the vertex coloring or edge coring of a graph.

This implies that χΛ(G) = χ(G) and χΛ1 (G) = χ1(G) if Λ = P2. But in general, the

Smarandachely Λ-coloring of a graph G is diff erent from that of its coloring. For exam-

ple, χP2 (Pn) = χP2

1= 2, χPk (Pn) = k , χ

Pk

1(Pn) = k − 1 for any integer 1 ≤ k ≤ n and a

Smarandachely P3-coloring on P7 can be found in Fig.3.1.9 following.

1 2 3 1 2 3 1

Fig.3.1.9

For the star S 1,n and circuit C n for integers 1 ≤ k ≤ n, we can easily find that

χPk (S 1,n) = 2 if k = 2,

n + 1 if k = 3,

1 if 4 ≤ k ≤ n,

χPk

1(S 1,n) =

1 if k = 2,

n if k = 3,

1 if 4 ≤ k ≤ n




and

χPk (C n) = χPk

1(C n) =

= mink + (i − 1) + si, 1 ≤ i ≤ n − k | n ≡ si(mod k + i − 1), 0 ≤ si < k + i − 1.

The following result is known by definition.

Theorem 3.1.16 Let H be a connected graph. Then

(1) χ H (nH ) = |V ( H )| and χ H 1 (nH ) = | E ( H )| , particularly, χG(G) = |V (G)| and

χG1

(G) = | E (G)|;(2) χ H (G) = χ H

1 (G) = 1 if H G.

Generally, we present the following problem.

Problem 3.1.1 For a graph G, determine the numbers χΛ(G) and χΛ1 (G) for subgraphs

Λ ≺ G.

Smarandachely Decomposition. Let P1 and P2 be graphical properties. A

Smarandachely (P1, P2)-decomposition of a graph G is a decomposition of G into sub-

graphs G1, G2, · · · , Gl ∈ P such that Gi ∈ P1 or Gi P2 for integers 1 ≤ i ≤ l.

If P1 or P2 = all graphs, a Smarandachely (P1, P2)-decomposition of a graph G

is said to be a Smarandachely P-decomposition. Particularly, if E (Gi)

∩E (G j)

≤k and

∆(Gi) ≤ d for integers 1 ≤ i, j ≤ l, such a Smarandachely P-decomposition is called a

Smarandache graphoidal (k , d)-cover of a graph G.

Furthermore, if d = ∆(G) or k = |G|, i.e., a Smarandachely graphoidal (k , ∆(G))-

cover with P = path or a Smarandachely graphoidal (k , ∆(G))-cover with P = treeis called a Smarandachely path k-cover or a Smarandache graphoidal tree d-cover of a

graph G for integers k , d ≥ 1. The minimum cardinalities of Smarandachely (P1, P2)-

decomposition and Smarandache graphoidal (k , d )-cover of a graph G are denoted by

ΠP1,P2(G), Π

(k ,d )

P (G), respectively.

Problem 3.1.3 For a graph G and properties P , P1 , P2 , determine ΠP1,P2(G) and

Π(k ,d )

P (G).

We only know partially results for Problem 3.1.3. For example,

Π(1,∆(G))

P (T ) = π(T ) =k

2



Sec.3.1 Graphs 91

for a tree T with k vertices of odd degree and

Π(1,∆(G))

P (W n) =

6 if n = 4,

n

2 + 3 if n

≥5

for a wheel W n = K 1 + C n−1 appeared in references [SNM1]-[SNM2].

Smarandachely Embeddable. Let T 1 and T 2 be two topological spaces. A graph

G is said to be Smarandachely (T 1, T 2)-embeddable into topological spaces T 1 and T 2 if

there exists a decomposition G = F

H 1

H 2, where F is a subgraph of G with a given

property P, H 1, H 2 are spanning subgraphs of G with two 1 − 1 continuous mappings

f : H 1 → T 1 and g : H 2 → T 2 such that f ( p) f (q) and g( p) g(q) if p, q V (G).

Furthermore, if

T 1 or

T 2 =

∅, i.e., a Smarandachely (

T ,

∅)-embeddable graph G is such a

graph embeddable in T if we remove a subgraph of G with a property P. Whence, we

know the following result for Smarandachely embeddable graphs by definition.

Theorem 3.1.17 Let T be topological space, G a graph and P a graphical property.

Then G is Smarandachely embedable in T if and only if there is a subgraph H ≺ G such

that G − H is embeddable in T .Particularly, if T is the Euclidean plane R2 and F a 1-factor, such a Smarandachely

embeddable graph G is called to be a Smarandachely planar graph. For example, al-

though the graph K 3,3 is not planar, but it is a Smarandachely planar graph shown inFig.3.1.10, where F = u1v1, u2v2, u3v3.

u1 u2 u3

v1v2 v3

Fig.3.1.10

Problem 3.1.4 Let T be a topological space. Determine which graph G is Smaran-

dachely T -embeddable.

The following result is an immediately consequence of Theorem 3.1.10.




Theorem 3.1.18 A graph G is Smarandachely planar if and only if there exists a 1-factor

F ≺ G such that there are no subgraphs homeomorphic to K 5 or K 3,3 in G − F.

§3.2 GRAPH GROUPS

3.2.1 Graph Automorphism. Let G1 and G2 be two isomorphic graphs. If G1 = G2 = G,

an isomorphism between G1 and G2 is called to be an automorphism of G. It should be

noted that all automorphisms of a graph G form a group under the composition operation,

i.e., φθ ( x) = φ(θ ( x)), where x ∈ E (G)

V (G). Such a graph is called the automorphism

group of G and denoted by AutG.

G AutG order

Pn Z 2 2

C n Dn 2n

K n S n n!

K m,n(m n) S m × S n m!n!

K n,n S 2[S n] 2n!2

Table 3.2.1

It can be immediately verified that AutG ≤ S n, where n = |G|. In Table 3.2.1, we

present automorphism groups of some graphs. But in general, it is very hard to present

the automorphism group AutG of a graph G.

3.2.2 Graph Group. Let (Γ; ) be a group. Then (Γ; ) is said to be a graph group if

there is a graph G such that (Γ, ) is isomorphic to a subgroup of AurG. Frucht proved

that for any finite group (Γ; ) there are always exists a graph G such that Γ AutG in

1938. Whence, the set of automorphism groups of graphs is equal to that of groups.

Let S ⊂ Γ with 1Γ S and S −1 = x−1| x ∈ S = S . A Cayley graph G = Cay(Γ : S )

of Γ on S ⊂ Γ is defined by

V (G) = Γ;

E (G) = (g, h)|g−1 h ∈ S .




Sec.3.2 Graph Group 93

Theorem 3.2.1 Let (Γ; ) be a finite group, S ⊂ Γ, S −1 = S and 1Γ S . Then L Γ ≤ Aut X,

where X = Cay(Γ : S ).

Proof For

∀g

∈Γ, we prove that the left representation τg : x

→g−1

x of g for

∀ x ∈ Γ is an automorphism of X . In fact, by

(g−1 x)−1 (g−1 y) = x−1 g g−1 y = x−1 y,

we know that

τg( x, y) = (τg( x), τg( y)),

i.e., τg ∈ Aut(Cay(G : S )). Whence, we get that L Γ ≤ Cay(Γ : S ).

A Cayley graph Cay(Γ : S ) is called to be normal if L Γ ⊳ Aut(Cay(G : S )), which

was introduced by Xu for the study of arc-transitive or half-transitive graphs in [Xum2].

The importance of this conception on Cayley graphs can be found in the following result.

Theorem 3.2.2 A Cayley graph Cay(Γ : S ) of a finite group (Γ; ) on S ⊂ Γ is normal if

and only if Aut(rmCay(Γ : S )) = L Γ Aut(Γ, S ) , where Aut(G, S ) = α ∈ AutΓ|S α = S .

Proof Notice that the normalizer of L Γ in the symmetric group S Γ is L Γ AutΓ. We

get that

N Aut(Cay(Γ:S ))(L Γ) = L Γ AutΓAut(Cay(Γ : S )) = L Γ (AutΓ A1Γ ).

That is N Aut(Cay(Γ:S ))(L Γ) = L Γ Aut(Γ, S ). Whence, Cay(Γ : S ) is normal if and only if

Aut(Cay(Γ : S )) = L Γ Aut(Γ, S ).

The following open problem presented by Xu in [Xum2] is important for finding the

automorphism group of a graph.

Problem 3.2.1 Determine all normally Cayley graphs for a finite group (Γ; ).

Today, we have know a few results partially answer Problem 3.2.1. Here we only list

some of them without proof. The first result shows that all finite groups have a normal

representation except for two special families.

Theorem 3.2.3([WWX1]) There is a normal Cayley graph for a finite group except for

groups Z 4 × Z 2 and Q8 × Z m2

for m ≥ 0.

For Abelian groups, we know the following result for the normality of Cayley graphs.








Certainly, an edge Γ-transitive graph G for any subgroup ∀Γ ≤ AutG must be an edge-

transitive graph. But the inverse is not always true. For example, the complete graph K n

for an integer n ≥ 3 is an edge-transitive graph with AutK n = S n. Let Γ = σ, where σ ∈

AutK n with σn

= 1S n . Then K n is not edge Γ-transitive since |Γ| = n <

n(n

−1)

2 = | E (K n)|.By Theorem 2.2.1, Γ can not be transitive on E (K n).

Γ-Action on Arc Set. Denoted by X (G) = (u, v)|uv ∈ E (G) the arc set of a graph

G. The Γ-action on X (G) is an action on X (G) induced by

ϕ( x, y) = (ϕ( x), ϕ( y)) ∈ X (G) for ∀( x, y) ∈ X (G)

for an automorphism ϕ ∈ Γ. Similarly, a graph G is called to be arc Γ-transitive for its a

graph group Γ if Γ is transitive on X (G), and to be arc-transitive if AutG is transitive on

X (G). The following result is obvious by definition.

Theorem 3.2.5 Any arc Γ-transitive graph G is an edge Γ-transitive graph. Conversely,

an edge Γ-transitive graph G is arc Γ-transitive if and only if there are involutions θ ∈ Γ

such that ( x, y)θ = ( y, x) for ∀( x, y) ∈ E (G).

§3.3 SYMMETRIC GRAPHS

3.3.1 Vertex-Transitive Graph. There are many vertex-transitive graphs. For example,

by Theorem 3.2.1 we know that all Cayley graphs is vertex-transitive.

Theorem 3.3.1 Any Cayley graph Cay(Γ : S ) on S ⊂ Γ is vertex-transitive.

Denoted by ( Z n; +) the additive group module n with Z n = 0, 1, 2, · · · , n − 1. A

circulant graph is a Cayley graph Cay( Z n : S ) for S ⊂ S n. Theorem 3.3.1 implies that

Cayley graphs are a subclass of vertex-transitive graphs. But if the order |V (G)| of a

vertex-transitive graph G is prime, Turner showed each of them is a Cayley graph, i.e.,

the following result in 1967.

Theorem 3.3.2 If G is a vertex-transitive graph of prime order p, then it is a circulant

graph.

Proof Let V (G) = u0, u1, · · · , u p−1 and H the stabilizer of u0. Suppose that σi ∈AutG is such an element that σi(u0) = ui. Applying Theorem 2.2.1, we get that |AutG| =



Sec.3.3 Symmetric Graphs 97

| H ||uAutG0

| = p| H |. Thus p||AutG|. By Sylow’s theorem, there is a subgroup K = 1, θ, · · · ,

θ p−1 of order p in AutG. Relabeling the vertices u0, u1, · · · , u p−1 by v0, v1, · · · , v p−1 so that

θ (vi) = vi+1 and θ (v p−1) = v0 for 0 ≤ i ≤ p−2. Suppose (v0, v1) ∈ E (G). Then by definition,

(vi, v2i) = (v0, vi)θ i

, (v2i, v3i) = (vi, v2i)θ i

, · · ·, (v( p−1)i, v0) = (v( p−2)i, v( p−1)i)θ i

∈ E (G). Thusv0viv2i · · · v( p−1)i forms a circuit in G. Now if we write vi as i and define S = i|(v0, vi) ∈ E (G), then G is nothing but the circulant graph Cay( Z p : S ).

It should be noted that not every every vertex-transitive graph is a Cayley graph . For

example, the Petersen graph shown in Fig.3.3.1 is vertex-transitive but it is not a Cayley

graph (See [Yap1] for details).

u1

u2

v1

u3u4

u5

v2

v3v4

v5

Fig.3.3.1

However, there is a constructing way shown in Theorem 3 .3.4 following such that every

vertex-transitive graph almost likes a Cayley graph, found by Sabidussi in 1964. For

proving this result, we need the following result first.

Theorem 3.3.3 Let H be a subgroup of a finite group (Γ; ) and S a subset of Γ with

S −1 = S , S ∩ H = ∅. If G is a graph with vertex set V (G) = Γ/H and edge set

E (G) = ( x H , y H )| x−1 y ∈ H S H , called the group-coset graph of Γ/H respect

to S and denoted by G(Γ/H : S ) , then G is vertex-transitive.

Proof First, we claim the graph G is well-defined. This assertion need us to show

that if ( x H , y H ) ∈ E (G) and x1 ∈ x H , y1 ∈ y H , then there must be

( x1 H , y1 H ) ∈ E (G). In fact, there are h, g ∈ H such that x1 = x h and y1 = y g

by definition. Notice that

x−1 y ∈ H S H ⇒ ( x h)−1 ( y g) ∈ H S H ⇒ x−11 y1 ∈ H S H .

Whence, ( x H , y H ) ∈ E (G) implies that ( x1 H , y1 H ) ∈ E (G).

Now for each g ∈ Γ, define a permutation φg on V (G) = Γ/H by φg( x H ) =




g x H for x H ∈ Γ/H . Then by

x−1 y ∈ H S H ⇒ (g x)−1 (g y) ∈ H S H ⇒ φ−1g ( x) φg( y) ∈ H S H ,

we find that ( xH , yH ) ∈ E (G) implies that (φg( x)H , φg( y)H ) ∈ E (G). Therefore,

φg is an automorphism of G.

Finally, for any x H , y H ∈ V (G), let g = y x−1. Then φg( x H ) = y x−1 ( x H ) = y H . Whence, G is vertex-transitive.

Now we can prove the Sabidussi’s representation theorem for finite groups following.

Theorem 3.3.4 Let G be a vertex-transitive graph and H = (AutG)u the stabilizer of a

vertex u ∈ V (G) with the composition operation . Then G is isomorphic with the group-

coset graph G(AutG/H : S ) , where S is the set of automorphisms σ of G such that (u, σ(u)) ∈ E (G).

Proof By definition, we are easily find that S −1 = S and S ∩ H = ∅. Define

π : AutG/H → G by π( x H ) = x(u), where x H ∈ Γ/H . We show that π is a

mapping. In fact, let x H = y H . Then there is h ∈ H such that y = x h. So

π( y H ) = y(u) = ( x h)(u) = x(h(u)) = x(u) = π( x ( H )).

Now we show that π is in fact a graph isomorphism following.

(1) π is 1 − 1. Otherwise, let π( x H ) = π( y ). Then x(u) = y(u) ⇒ y−1 x(u) =

u ⇒ y−1 x ∈ H ⇒ y ∈ x H ⇒ x H = y H .

(2) π is onto. Let v ∈ V (G). Notice that G is vertex-transitive. There exists z ∈ AutG

such that z(u) = v, i.e., π( z H ) = z(u) = v.

(3) π preserves adjacency in G. By definition, ( xH , yH ) ∈ E (G(AutG/H , S )) ⇔ x−1 y ∈ H S H ⇔ x−1 y = h z g for some h, g ∈ H , z ∈ S ⇔ h−1 x−1 y g−1 =

z ⇔ (u, h−1 x−1 y g−1(u)) ∈ E (G) ⇔ (u, x−1 y(u)) ∈ E (G) ⇔ ( x(u), y(u)) ∈ E (G) ⇔

(π( x H ), π( y H )) ∈ E (G).Combining (1)-(3), the proof is completes.

Theorem 3.3.4 enables one to know which vertex-transitive graph G is a Cayley

graph. By Theorem 2.1.1(2), we know that any two stabilizers (AutG)u, (AutG)v for u, v ∈V (G) are conjugate in AutG. Consequently, (AutG)u is normal if and only if (AutG)u =

1AutG. By definition, the group-coset graph G(AutG/H : S ) in Theorem 3.3.4 is a




Cayley graph if and only if AutG/H is a quotient group. But this just means that H ⊳

AutG by Theorem 1.3.2. Combining these facts, we get the necessary and sufficient

condition for a vertex-transitive graph to be a Cayley graph by Theorem 3 .3.4 following.

Theorem 3.3.5 A vertex-transitive graph G is a Cayley graph if and only if the action of

AutG on V (G) is regular.

Generally, let (Γ; ) be a finite group. A graph G is called to be a graphical regular

representation (GRR) of Γ if AutG Γ and AutG acts regularly transitive on V (G). Such

a group Γ is called to have a GRR. We needed to answer the following problem.

Problem 3.3.1 Determine each finite group Γ with a GRR.

A simple case for Problem 3.3.1 is finite Abelian groups. We know the followingresult due to Chao and Sabidussi in 1964.

Theorem 3.3.6 Let G be a graph with an Abelian automorphism group AutG acts transi-

tively on V (G). Then AutG acts regularly transitive on V (G) and AutG is an elementary

Abelian 2-group.

Proof According to Theorem 2.2.2, we know that AutG acts regularly transitive

on V (G). Now since AutG acts regularly on V (G), G is isomorphic to a Cayley graph

Cay(AutG : S ). Because AutG is Abelian, τ : g → g−1

is an automorphism of AutGand fixes S setwise. It can be shown that this automorphism is an automorphism of

Cay(AutG : S ) fixing the identity element of AutG. Whence, g = τ(g) = g−1 by the fact

of regularity for every g ∈ AutG. So AutG is an elementary 2-groups.

Theorem 3.3.6 claims that an Abelian group Γ has a GRR only if Γ = Z n2 for some

integers n ≥ 1. In fact, by the work of McAndrew in 1965, we know a complete answer

for Problem 3.3.1 in this case following.

Theorem 3.3.7 An Abelian group Γ has a GRR if and only if Γ = Z n2

for n = 1 or n≥

5.

A generalized dicylic group (Γ; ) is a non-Abelian group possing a subgroup (H ; )

of index 2 and an element γ of order 4 such that γ −1hγ = h−1 for ∀h ∈ H . By following

the work of Imrich, Nowitz, Watkins, Babai, etc., Hetzel and Godsil respective answered

Problem 3.3.1 for solvable groups and non-solvable groups. They get the following result

(See [God1]-[God2] and [Cam1] for details) independently.




Theorem 3.3.8 A finite group (Γ; ) possesses no GRR if and only if it is an Abelian group

of exponent greater than 2 , a generalized dicyclic group, or one of 13 exceptional groups

following:

(1) Z 22 , Z 32 , Z 42 ;

(2) D6, D8, D10;

(3) A4;

(4)

a, b, c|a2 = b2 = c2 = 1Γ, a b c = b c a = c a b

;

(5)

a, b|a8 = b2 = 1Γ, b a b = b5

;

(6)

a, b, c|a3 = b2 = c3 = (a b)2 = (c b)2 = 1Γ, a c = c a

;

(7)

a, b, c|a3 = b3 = c3 = 1Γ, a c = c a, b c = c b, c = a−1 b−1 a b

;

(8) Q8

× Z 3, Q8

× Z 4.

3.3.2 Edge-Transitive Graph. Certainly, the edge-transitive graphs are closely related

with vertex-transitive graphs by definition. We can easily obtain the following result.

Theorem 3.3.9 Let G be an edge-transitive graph without isolated vertices. Then

(1) G is vertex-transitive, or

(2) G is bipartite with two vertex-orbits under the action AutG on V (G) to be its

vertex bipartition.

Proof Choose an edge e = uv ∈ E (G). Denoted by V 1 and V 2 the orbits of u andv under the action of AutG on V (G). Then we know that V 1 ∪ V 2 = V (G) by the edge-

transitivity of G. Our discussion is divided into toe cases following.

Case 1. If V 1 ∩ V 2 ∅, then G is vertex-transitive.

Let x and y be any two vertices of G. If x, y ∈ V 1 or x, y ∈ V 1, for example, x, y ∈ V 1,

then there exist σ, ς ∈ AutG such that σ(u) = x and ς (u) = y. Thus ςσ−1 is such an

automorphism with ςσ−1( x) = y. If x ∈ V 1 and y ∈ V 2, let w ∈ V 1 ∩ V 2. By assumption,

there are φ, ϕ

∈AutG such that φ( x) = ϕ( y) = w. Then we get that ϕ−1φ( x) = y, i.e., G is

vertex-transitive.

Case 2. If V 1 ∩ V 2 = ∅, then G is bipartite.

Let x, y ∈ V 1. If xy ∈ E (G), then there are σ ∈ AutG such that σ(uv) = xy. But this

implies that one of x, y in V 1 and another in V 2, a contradiction. Similarly, if x, y ∈ V 2,

then xy E (G). Whence, G is a bipartite graph.




We get the following consequences by this result.

Corollary 3.3.1 Let G be a regular edge-transitive graph with an odd degree d ≥ 1. If

|G

| ≡1(mod2) , then G is vertex-transitive.

Proof Notice that if G is bipartite, then |V 1|d = |V 2|d = ε(G). Whence, |G| =

|V 1| + |V 2| ≡ 0(mod2), a contradiction.

Corollary 3.3.2 Let G be a regular edge-transitive graph of degree d ≥ |G|/2. Then G is

vertex-transitive.

u1 u2

u3u4

u5 u6

Fig.3.3.2

In fact, there are many edge-transitive but not vertex-transitive graphs, and vertex-transitive

but not edge-transitive graphs. For example, the complete graph K n1,n2with n1 n2 is

edge-transitive but not vertex-transitive, and the graph shown in Fig.3.3.2 is a vertex-

transitive but not edge-transitive graph.

3.3.3 Arc-Transitive Graph. An s-arc of a graph G is a sequence of vertices v0, v1, · · · , vs

such that consecutive vertices are adjacent and vi−1 vi+1 for 0 < i < s. For example, a

circuit C n is s-arc transitive for all s ≤ n. A graph G is s-arc transitive if AutG is transitive

on s-arcs. For s ≥ 1, it is obvious that an s-arc transitive graph is also (s−1)-arc transitive.

A 0-arc transitive graph is just the vertex-transitive, and a 1-arc transitive graph is usually

called to be arc-transitive graph or symmetric graph.

Tutte proved the following result for s-arc transitive cubic graphs in 1947 (See in

[Yap1] for its proof).

Theorem 3.3.10 Let G be a s-arc transitive cubic graph. Then s ≤ 5.

Examples of s-arc transitive cubic graphs for s ≤ 5 can be found in [Big2] or [GoR1].

Now we turn our attention to symmetric graphs.

Let Z p = 0, 1, · · · , p − 1 be the cyclic group of order p written additively. We know




that Aut Z p is isomorphic to Z p−1. For a positive divisor r of p−1, let H r denote the unique

subgroup of Aut Z p of order r , H r ≃ Z r . Define a graph G( p, r ) of order p by

V (G( p, r )) = Z p, E (G( p, r )) = xy| x − y ∈ H r .

A classification of symmetric graph with a prime order p was obtained by Chao. He

proved the following result in 1971.

Theorem 3.3.11 Let p be an odd prime. Then a graph G of order p is symmetric if and

only if G = pK 1 or G = G( p, r ) for some even divisor r of p − 1.

In the reference [PWX1] and [WaX1], we can also find the classification of symmet-

ric graphs of order a product of two distinct primes. For example, there are 12 classes

of symmetric graphs of order 3 p, where p > 3 is a prime, including 3 pK 1, pK 3, 3G( p, r )

for an even divisor r of p − 1, G(3 p, r ) for a divisor of p − 1, G( p, r )[3K 1], K 3 p and

other 6 classes, where G(3 p, r ) is defined by V (G(3 p, r )) = xi | i ∈ Z 3, x ∈ Z p and

E (G(3 p, r )) = ( xi, yi+1) | i ∈ Z 3, x, y ∈ Z p and y − x ∈ H r .

A graph G is half-transitive if G is vertex-transitive and edge-transitive, but not arc-

transitive. Tuute found the following result.

Theorem 3.3.12 If a graph G is vertex-transitive and edge-transitive with a odd valency,

then G must be arc-transitive.

Proof Let uv ∈ E (G). Then we get two arcs (u, v) and (v, u). Define Ω1 = (u, v)AutG =

(u, v)g|g ∈ AutG and Ω2 = (v, u)AutG = (v, u)g|g ∈ AutG. By the transitivity of AutG

on E (G), we know that Ω1 ∪ Ω2 = A(G), where A(G) denote the arc set of G. If G is

not arc-transitive, there must be Ω1 ∩ Ω2 = ∅. Namely, there are no g ∈ AutG such that

( x, y)g = ( y, x) for ∀( x, y) ∈ A(G). Now let X v = x|(v, x) ∈ Ω1 and Y v = y|( y, v) ∈ Ω1.

Then X v ∩ Y v = ∅. Whence, N G(v) = X v ∪ Y v. This fact enables us to know the valency

of G is k = | X v| + |Y v|. By the transitivity of AutG on V (G), we know that | X v| = | X u| and

|Y v| = |Y u| for ∀u ∈ V (G). So | E (G)| = | X v||V (G)| = |Y v||V (G)|. We get that | X v| = |Y v|, i.e.,k is an even number, a contradiction.

By Theorem 3.3.12, a half-transitive graph must has even valency. In 1970, Bouwer

constructed half-transitive graphs of valency k for each even number k > 2 and the mini-

mum half-transitive graph is a 4-regular graph with 27 vertices found by Holt in 1981. In

1992, Xu proved this minimum half-transitive graph is unique (See [XHLL1] for details).



Sec.3.4 Graph Semi-Arc Groups 103

§3.4 GRAPH SEMI-ARC GROUPS

3.4.1 Semi-Arc Set. Let G be a graph, maybe with loops and multiple edges, e = uv ∈ E (G). We divide e into two semi-arcs e+

u , e−u (or e+

u , e+v ), and call such a vertex u to be the

root vertex of e+u . Here, we adopt a convention following:

Convention 3.4.1 Let G be a graph. Then for e = uv ∈ E (G) , e−u = e+

v if u v,

e−u e+

v if u = v.

Denote by X 12

(G) the set of all such semi-arcs of a graph G. We present a few

examples for X 12

(G). Let D0.3.0, B3,K 4 be the dipole, bouquet and the complete graph

shown in Fig.3.4.1.

D0,3,0 B3 K 4

u v

O

u1 u2

u3 u4

e1

e2

e3

e1

e2

e3

Fig.3.4.1

Then, we know their semi-arc sets as follows:

X 12

( D0.3.0) = e1+u , e2+

u , e3+u , e1+

v , e2+v , e3+

v ,

X 12

( B3) = e1+O , e2+

O , e3+O , e1−

O , e2−O , e3−

O ,

X 12

(K 4) = u1u+2 , u1u−

2 , u1u+3 , u1u−

3 , u1u+4 , u1u−

4 , u2u+3 , u2u−

3 , u2u+4 , u2u−

4 , u3u+4 , u3u−

4 .

Notice that the Convention 3.4.1 and these examples show that we can represent all

semi-arcs of a graph G by elements in V (G) ∪ E (G) ∪ +, − in general, and all semi-arcs

of G can be represent by elements in V (G) ∪ E (G) ∪ + or by elements in V (G) ∪ +if and only if G is a graph without loops, or neither with loops or multiple edges, i.e., a

simple graph G.

Two semi-arc eu, f •v with , • ∈ +, − are said incident if u = v, e f with = • =




+, or e = f , u v with = •, or e = f , u = v with = +, • = −. For example, e2+u and

e2+v in D0.3.0, e2+

O and e2−O in B3 in Fig.3.4.1 both are incident.

3.4.2 Graph Semi-Arc Group. We have know the conception of automorphism of a

graph in Section 3.1. Generally, an automorphism of a graph G on V (G) E (G) is an

1 − 1 mapping ( ξ,η) on G such that

ξ : V (G) → V (G), η : E (G) → E (G)

satisfying that for any incident elements e, f , ( ξ,η)(e) and ( ξ,η)( f ) are also incident. Cer-

tainly, all such automorphisms of a graph G also form a group, denoted by AutG.

We generalize this conception to that of the semi-arc set X 12

(G). The semi-arc auto-

morphism of a graph was first appeared in [Mao1], and then applied for the enumeration

maps on surfaces underlying a graph Γ in [MaL3] and [MLW1], which is formally defined

following.

Definition 3.4.1 Let G be a graph. A 1 − 1 mapping ξ on X 12

(G) is called a semi-arc

automorphism of the graph G if for ∀eu, f •v ∈ X 1

2(G) with , • ∈ +, − , ξ (e

u) and ξ ( f •v ) are

incident if and only if eu and f •v are incident.

By Definition 3.4.1, all semi-arc automorphisms of a graph form a group under the

composition operation, denoted by Aut 12

G, which is important for the enumeration of

maps on surfaces underlying a graph and determining the conformal transformations on aKlein surface.

The Table 3.4.1 following lists semi-arc automorphism groups of a few well-known

graphs.

G Aut 12

G order

K n S n n!

K m,n(m n) S m × S n m!n!

K n,n S 2[S n] 2n!2

Bn S n[S 2] 2nn!

D0.n.0 S 2 × S n 2n!

Dn.k .l(k l) S 2[S k ] × S n × S 2[S l] 2k +ln!k !l!

Dn.k .k S 2 × S n × (S 2[S k ])2 22k +1n!k !2

Table 3.4.1



Sec.3.4 Graph Semi-Arc Groups 105

In this table, D0.n.0 is a dipole graph with 2 vertices, n multiple edges and Dn.k .l is a

generalized dipole graph with 2 vertices, n multiple edges, and one vertex with k bouquets

and another, l bouquets. This table also enables us to find some useful information for

semi-arc automorphism groups. For example, Aut 12 K n = AutK n = S n, Aut 1

2 Bn = S n[S 2]but Aut Bn = S n, i.e., Aut 1

2 Bn Aut Bn for any integer n ≥ 1.

Comparing semi-arc automorphism groups in Table 3, 4, 1 with that of Table 3.2.1, it

is easily to find that the semi-arc automorphism group are the same as the automorphism

group in the first two cases. Generally, we know a result related the semi-arc automor-

phism group with that of automorphism group of a graph, i.e., Theorem 3.4.1 following.

For this objective, we introduce a few conceptions first.

For

∀g

∈AutG, there is an induced action g

|

12 on X 1

2(G) , g : X 1

2(G)

→X 1

2(G)

determined by

∀eu ∈ X 12

(G), g(eu) = (g(e)g(u).

All induced action of the elements in AutG on X 12

(G) is denoted by AutG| 12 . Notice that

AutG AutG| 12 . We get the following result.

Theorem 3.4.1 Let G be a graph without loops. Then Aut 12

G = AutG| 12 .

Proof By the definition, we only need to prove that for ∀ ξ 12

∈ Aut 12

G, ξ = ξ 12|G ∈

AutG and ξ 12

= ξ | 12 . In fact, Let e

u, f • x ∈ X 12

(G) with , • ∈ +, −, where e = uv ∈ E (G),

f = xy ∈ E (G). Now if

ξ 12

(eu) = f • x ,

by definition, we know that ξ 12

(ev) = f • y . Whence, ξ 1

2(e) = f . That is, ξ 1

2|G ∈ AutG.

By assumption, there are no loops in G. Whence, we know that ξ 12|G = 1AutG if and

only if ξ 12

= 1Aut 12

G. So ξ 12

is induced by ξ 12|G on X 1

2(G). Thus,

Aut 12

G = AutG| 12 .

We have know that Aut 12 Bn Aut Bn for any integer n ≥ 1. Combining this fact with

Theorem 3.4.1, we know the following.

Theorem 3.4.2 Let G be a graph. Then Aut 12

G = AutG| 12 if and only if G is a loopless

graph.

3.4.3 Semi-Arc Transitive Graph. A graph G is called to be semi-arc transitive if

Aut 12

G is action transitively on X 12

(G). For example, each of K n, Bn−1 and D0.n.0 for any




integer n ≥ 2 is semi-arc transitive. We know the following result for semi-arc transitive

graphs.

Theorem 3.4.3 A graph G is semi-arc transitive if and only if it is arc-transitive.

Proof A semi-arc transitive graph G is arc-transitive by the definition of its preserv-

ing incidence of semi-arcs.

Conversely, let G be an arc-transitive graph. Let e+u and f +v ∈ X 1

2(G) with e = (u, x)

and f = (v, y). By assumption, G is arc-transitive. Consequently, there is an automor-

phism ς ∈ AutG such that ς (u, x) = (v, y). Then it is easily to know that ς (e+u ) = f +v , i.e.,

G is semi-arc transitive.

§3.5 GRAPH MULTIGROUPS

3.5.1 Graph Multigroup. There is a natural way for getting multigroups on graphs. Let

G be a graph, H ≺ G and σ ∈ AutG. Consider the localized action σ| H of σ on H . In

general, this action must not be an automorphism of H . For example, let G be the graph

shown in Fig.3.5.1 and H = v1, v2, v3G.

v1

v2

v3 v4

v5

v6

Fig.3.5.1

Let σ1 = (v1, v3)(v4, v6)(v2)(v5) and σ2 = (v1, v6)(v2, v5)(v3, v4). Then it is clear that

σ1, σ2 ∈ AutG and

H σ2 = v1, v2, v3G = H and H σ1 = v4, v5, v6G H .

Whence, σ1 is an automorphism of H , but σ2 is not. In fact, let ∀ς ∈ (AutG) H . Then

H ς = H , i.e., ς | H is an automorphism of H . Now define

AutG H = ς | H | ς ∈ (AutG) H .

Then AutG H is an automorphism group of H .



Sec.3.5 Graph Multigroups 107

An extended action g|G for an automorphism g ∈ Aut H i is the action of g on G by

introducing new actions of g on G \ V ( H i), 1 ≤ i ≤ m. The previous discussion enables

one to get the following result.

Theorem 3.5.1 Let G be a graph and G =m

i=1

H i a decomposition of G. Then for any

integer i, 1 ≤ i ≤ m, there is a subgroup Pi ≤ Aut H i such that Pi|G ≤ AutG, i.e.,P =m

i=1Pi is a multigroup.

Proof Choose Pi = AutG H i for any integer i, 1 ≤ i ≤ m. Then the result follows.

For a given decomposition G =m

i=1

H i of a graph G, we can always get automorphism

multigroups AutmulG =m

i=1

H i, H i ≤ Aut H i for integers 1 ≤ i ≤ m, which must not be

an automorphism group of G. For its dependence on the structure of G =m

i=1

H i, such

a multigroup AutmulG is denoted bym

i=1

H i in this book. Generally, the automorphism

multigroups of a graph G are not unique unless G = K 1. The maximal automorphism

multigroup of a graph G is AutmulG =m

i=1

Aut H i and the minimal is that of AutmulG =

mi=1

1Aut H i. We first determine automorphism groups of G in these multigroups following.

Let G be a graph, H ≺ G and σ ∈ Aut H , τ ∈ Aut(G \ V ( H )). They are called to be

in coordinating with each other if the mapping g : G → G determined by

g(v) =

σ(v), if v ∈ V ( H ),

τ(v), if v ∈ G \ V ( H )

is an automorphism of G for ∀v ∈ V (G). If such a g exists, we say τ can be extended to

G and denoted g by τG. Denoted by AutG H = σG |σ ∈ Aut H . Then it is clear that

AutG H = AutG H | H ≺ Aut H . We find the following result for the automorphism group of

a graph.

Theorem 3.5.2 Let G be a graph and H

≺G. Then the mapping φG : AutG

→Aut H

determined by φG(g) = g| H is a homomorphism, i.e., AutG/KerφG ≃ AutG H .

Proof For any automorphism g ∈ AutG, by Theorem 3.5.1, there is a localized action

g| H such that H g = H , g = g| H ∈ AutG H , i.e., such a correspondence φG is a mapping. We

are needed to prove the equality φG(ab) = φG(a)φG(b) holds for ∀a, b ∈ AutG. In fact,

φG(a)φG(b) = a|G H b|G H = (ab)|G H = φG(ab)




by the property of automorphism. Whence, φG is a homomorphism. Applying the homo-

morphism theorem of groups, we get AutG/KerφG ≃ KerφG. Notice that KerφG = AutG H .

We finally get that AutG/KerφG ≃ AutG H .

If φG is onto or 1−1, then KerφG = 1AutG or Aut H . We get the following consequence

by Theorem 3.5.2.

Corollary 3.5.1 Let G be a graph and H ≺ G. If the homomorphism φ : AutG → Aut H

is onto or 1 − 1 , then AutG/Kerφ ≃ Aut H or AutG ≃ AutG H .

For example, Let G be the graph shown in Fig.3.5.1 and H = v1, v3, v4, v6G. Then

σ1| H = (v1, v3)(v4, v6) and σ2| H = (v1, v6)(v3, v4), i.e., the homomorphism φG : AutG →AutG H is 1 − 1 and onto. Whence, we know that

AutG ≃ AutG H = σ1| H , σ2| H .

Although it is very difficult for determining the automorphism group of a graph G in

general, it is easy for that of automorphism multigroups if the decomposition G =m

i=1

H i

is chosen properly. The following result is easy obtained by definition.

Theorem 3.5.3 For any connected graph G,

Aut E G = (u,v)∈ E (G)

S u,v

is an automorphism multigroup of G, where S u,v is the symmetric group action on the

vertices u and v.

Proof Certainly, any graph G has a decomposition G =

(u,v)∈ E (G)

(u, v). Notice that

the automorphism on each edge (u, v) ∈ E (G) is that symmetric group S u,v. Then the

assertion is followed.

The automorphism multigroup Aut E G is a graphical property by Theorem 3.5.3.

Furthermore, we know that Aut E G is a graph invariant on G by the following result.

Theorem 3.5.4 Let G, H be two connected graphs. Then G is isomorphic to H if and

only if Aut E G and Aut E H are permutation equivalent, i.e., there is an isomorphism ς :

Aut E G → Aut E H and a 1 − 1 mapping ι : E (G) → E ( H ) such that ς (g)(ι(e)) = ι(g(e)) for

∀g ∈ AutG and e ∈ E (G).




Proof If G ≃ H , we are easily getting an isomorphism σ : V (G) → V ( H ), which

induces an isomorphism ς : Aut E G → Aut E H and a 1 − 1 mapping ι : E (G) → E ( H ) by

σ(u, v) = (σ(u), σ(v)) for ∀e = (u, v) ∈ E (G).

Now if there is an isomorphism ς : Aut E G → Aut E H and a 1−1 mapping ι : E (G) → E ( H ) such that ς (g)(ι(e)) = ι(g(e)) for ∀g ∈ AutG and e ∈ E (G), by definition

Aut E G =

(u,v)∈ E (G)

S u,v,

we know that

ς :

(u, x)∈ E (G) for x∈V (G)

S u, x →

(v, y)∈ E ( H ) for y∈V ( H )

S v, y,

where ι : (u, x)∈

E (G)→

(v, y)∈

E ( H ). Whence, ς and ι induce a 1−

1 mapping

σ :


(u, x) →

(v, y)∈ E ( H ) for y∈V ( H )

(v, y).

This fact implies that σ : u ∈ V (G) → v ∈ V ( H ) if we represent the vertices u, v re-

spectively by those of u


(u, x) and v

(v, y)∈ E ( H ) for y∈V ( H )

(v, y) in graphs

G and H , where the notation a b means the definition of a by that of b. Essentially,

such a mapping σ : V (G) → V ( H ) is an isomorphism between graphs G and H for easily

checking that

σ(u, x) = (σ(u), σ( x))

for ∀(u, x) ∈ E (G) by such representation of vertices in a graph. Thus G ≃ H .

The decomposition G =

(u,v)∈ E (G)

(u, v) is a K 2-decomposition. A clique decomposi-

tion of a graph G is such a decomposition G =m

i=1

K ni, where K ni

is a maximal complete

subgraph in G for integers 1 ≤ i ≤ m. We have know AutK ni= S ni

from Table 3.2.1.

Whence, we know the following result on automorphism multigroups of a graph.

Theorem 3.5.5 Let G =m

i=1

K nibe a clique decomposition of a graph G. Then AutmulG =

mi=1

H i is an automorphism multigroup of G, where H i ≤ S V (K ni).

Proof Notice that AutK ni= S ni

. Whence, AutmulG =m

i=1

H i is an automorphism

multigroup of G for each H i ≤ S V (K ni).






Example 3.5.1 Let P = (1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) be a permutation group

action on Ω = 1, 2, 3, 4. Then the action graph G[P ; Ω] is the complete graph K 4 with

labels shown in Fig.3.5.4,

1

2

34

α

α

β

β

γ

γ

Fig.3.5.4

in where α = (1, 2)(3, 4), β = (1, 3)(2, 4) and γ = (1, 4)(2, 3).

Example 3.5.2 Let P be a permutation multigroup action on Ω with

P = P1

P2 and Ω = 1, 2, 3, 4, 5, 6, 7, 8

1, 2, 5, 6, 9, 10, 11, 12,

where P1 = (1, 2, 3, 4), (5, 6, 7, 8) and P2 = (1, 5, 9, 10), (2, 6, 11, 12). Then the ac-

tion graph G[

P ; Ω] of

P on

Ω = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 is shown in Fig.3.5.5,

in where labels on edges are removed. It should be noted that this action graph is in facta union graph of four complete graphs K 4 with intersection vertices.

23

14 5

6 7

8

910

1112

Fig.3.5.5

These Examples 3.5.1 and 3.5.2 enables us to find the following result on the action

graphs of multigroups.




Theorem 3.5.7 Let P be a multigroup action on a set Ω with

P =

m

i=1

Pi and

Ω =

m

i=1

Ωi,

where each permutation group Pi acts on Ωi with orbits Ωi1, Ωi2, · · · , Ωisifor each integer

i, 1 ≤ i ≤ m. Then

G[ P ;Ω] =

mi=1

si j=1

K |Ωi j |

with intersections K |Ωi j∩Ωkl | only if for integers 1 ≤ i k ≤ m, 1 ≤ j ≤ si , l ≤ l ≤ sk .

Particularly, if m = 1 , i.e., P is just a permutation group, then its action graph G[ P ;Ω]

is a union of complete graphs without intersections.

Proof Notice that for each orbit Ωi j

of Pi

action on Ωi, the subgraph of the action

graph is the complete graph K |Ωi j | and Ωi j1∩ Ωi j2

= ∅ if j1 j2, i.e.,, K |Ωi j1| ∩ K |Ωi j2

| = ∅.

This result follows by definition.

By Theorem 3.5.5, we are easily find the automorphism groups of the graph shown

in Fig.3.5.5, particularly the maximal automorphism group following:

AutclG[ P; Ω] = S 1,2,3,4

S 5,6,7,8

S 1,5,9,10

S 2,6,11,12.

Generally, we get the following result.

Theorem 3.5.8 Let P be a multigroup action on a set Ω with P =m

i=1Pi and Ω =

mi=1

Ωi,

where each permutation group Pi acts on Ωi with orbits Ωi1, Ωi2, · · · , Ωisifor each integer

i, 1 ≤ i ≤ m. Then the maximal automorphism group of G[ P ;Ω] is

AutclG[ P; Ω] =

mi=1

si j=1

S Ωi j.

Particularly, if |Ωi j ∩ Ωkl| = 1 for i k, 1 ≤ i, k ≤ m, 1 ≤ j ≤ si , l ≤ l ≤ sk , then

Autcl

G[ P ;Ω] =

m

i=1

si

j=1

SΩi j

.

Proof Notice that if |Ωi j ∩ Ωkl| = 1 for i k , 1 ≤ i, k ≤ m, 1 ≤ j ≤ si, l ≤ l ≤ sk , then

G[ P; Ω] =

mi=1

si j=1

K |Ωi j |.

This result follows from Theorems 3.5.5 and 3.5.7.




3.5.3 Globally Transitivity. Let P be a permutation multigroup action on Ω. This

permutation multigroup P is said to be globally k-transitive for an integer k ≥ 1 if for

any two k -tuples x1, x2, · · · , xk ∈ Ωi and y1, y2, · · · , yk ∈ Ω j, where 1 ≤ i, j ≤ m, there are

permutations π1, π2, · · · , πn ∈ P such that xπ1π2

···πn

1 = y1, xπ1π2

···πn

2 = yi, · · · , xπ1π2

···πn

k = yk .We have obtained Theorems 2.6.8-2.6.10 for characterizing the globally transitivity of

multigroups. In this subsection, we characterize it by the action graphs of multigroups.

First, we know the following result on globally 1-transitivity, i.e., the globally transitivity

of a multigroup.


P =

m

i=1

Pi and Ω =

m

i=1

Ωi,

where each permutation group Pi acts on Ωi for integers 1 ≤ i ≤ m. Then P is globally

transitive action on Ω if and only if G[ P; Ω] is connected.

Proof Let x, y ∈ Ω. If P is globally transitive action on Ω, then there are elements

π1, π2, · · · , πn ∈ P such that xπ1π2···πn = y for an integer n ≥ 1. Define v1 = xπ1 , v2 =

xπ1π2 , · · · , vn−1 = xπ1π2···πn−1 . Notice that v1, v2, · · · , vn−1 ∈ Ω. By definition, we conse-

quently find a walk (path) xv1v2 · · · vn−1 y in the action graph G[

P;

Ω] for any two vertices

x, y

∈V (G[ P ;Ω]), which implies that G[ P ;Ω] is connected.

Conversely, if G[ P ;Ω] is connected, for ∀ x, y ∈ V ((G[ P; Ω])) = Ω, let xu1 · · · un−1 y

be a shortest path connected the vertices x and y in G[ P;Ω] for an integer n ≥ 1. By

definition, there are must be π1, π2, · · · , πn ∈ P such that xπ1 = u1, uπ2

1= u2, · · · , uπn

n−1= y.

Whence,

xπ1π2···πn = y.

Thus P is globally transitive action on Ω.

For a multigroup action

P action on

Ω with

P =

mi=1

Pi and Ω =

mi=1

Ωi,

where each permutation group Pi acts on Ωi for integers 1 ≤ i ≤ m, define

Ωk i = ( x1, x2, · · · , xk ) | xl ∈ Ω and Ωk =

mi=1

Ωk i




for integers k ≥ 1 and 1 ≤ i ≤ m. Then we are easily proved that a permutation group

P action on Ω is k-transitive if and only if P action on Ωk is transitive for an integer

k ≥ 1. Combining this fact with that of Theorem 3.5.9, we get the following result on the

globally k -transitivity of multigroups.


P =

mi=1

Pi and Ω =

mi=1

Ωi,

where each permutation group Pi acts on Ωi for integers 1 ≤ i ≤ m. Then P is globally

k -transitive action on

Ω for an integer k ≥ 1 if and only if G[

P ;

Ωk ] is connected.

Proof Replacing Ω by Ωk

in the proof of Theorem 3.5.9 and applying the fact that apermutation group P action on Ω is k -transitive if and only if P action on Ωk is transitive

for an integer k ≥ 1, we get our conclusion.

Applying the action graph G[ P ;Ω] and G[ P ;Ωk ], we can also characterize the

globally primitivity or other properties of permutation multigroups by graph structure.

All of those are laid the reader as exercises.

§3.6 REMARKS

3.6.1 For catering to the need of computer science, graphs were out of games and turned

into graph theory in last century. Today, it has become a fundamental tool for dealing

with relations of events applied to more and more fields, such as those of algebra, topol-

ogy, geometry, probability, computer science, chemistry, electrical network, theoretical

physics, · · · and real-life problems. There are many excellent monographs for its theo-

retical results with applications, such as these references [ChL1], [Whi1] and [Yap1] for

graphs with structures, [GrT1], [MoT1] and [Liu1] for graphs on surfaces.

3.6.2 The conception of Smarandachely graph property in Subsection 3.1.4 is presented

by Smarandache systems or Smarandache’s notion, i.e., such a mathematical system in

which there is a rule that behaves in at least two diff erent ways, i.e., validated and in-

valided, or only invalided but in multiple distinct ways (See [Mao2]-[Mao4], [Mao25]

and [Sma1]-[Sma2] for details). In fact, there are two ways to look a graph with more



Sec.3.6 Remarks 115

than one edges as a Smarandachely graph. One is by its graphical structure. Another

is by graph invariants on it. All of those Smarandachely conceptions are new and open

problems in this subsection are valuable for further research.

3.6.3 For surveying symmetries on graphs, automorphisms are needed, which is permu-

tations on graphs. This is the closely related place of groups with that of graphs. In fact,

finite graphs are a well objectives for applying groups, particularly for classifying sym-

metric graphs in recent two decades. To determining the automorphism groups AutG of

a graph G is an important but more difficult problem, which enables one to enumerating

maps on surfaces underlying G, or find regular maps on surfaces (See following chapters

in this book). Sections 3.2-3.3 present two ways already known. One is the GRR of finite

group. Another is the normally Cayley graphs for finite groups. More results and exam-

ples can be found in references [Big2], [GoR1], [Xum2], [XHL1] and [Yap1] for further

reading.

3.6.4 A hypergraph Λ is a triple (V , f , E ) with disjoints V , E and f : E → P(V ),

where each element in V is called the vertex and that in E is called the edge of Λ. If

f : E → V × V , then a hypergraph Λ is nothing but just a graph G. Two elements

x ∈ V , e ∈ E of a hypergraph (V , f , E ) are called to be incident if x ∈ f (e). Two hy-

pergraphs Λ1 = (V 1, f 1, E 1) and Λ1 = (V 2, f 2, E 2) are isomorphic if there exists bijections

p : E 1 → E 2, q : V 1 → V 2 such that q[ f 1(e)] = f 2( p(e)) holds for ∀e ∈ E . Particularly,if Λ1 = Λ2, i.e., isomorphism between a hypergraph Λ, such an isomorphism is called

an automorphism of Λ. All automorphisms of a hypergraph Λ form a group, denoted

by AutΛ. For hypergraphs, we can also introduce conceptions such as those of vertex-

transitive, edge-transitive, arc-transitive, semi-arc transitive and primitive by the action of

AutΛ on Λ and get results for symmetric hypergraphs. As we known, there are nearly

none such results found in publication.

3.6.5 The semi-arc automorphism of a graph is firstly introduced in [Mao1] and [Mao2]

for enumerating maps on surfaces underlying a graph. Besides of these two references,

further applications of this conception can be found in [Mao5], [MaL3], [MLW1] and

[Liu4]. It should be noted that the semi-arc automorphism is called semi-automorphism

of a graph in [Liu4]. In fact, the semi-arc automorphism group of a graph G is the induced

action of AutG on semi-arcs of G if G is loopless. Thus is the essence of Theorems 3.4.1

and 3.4.2. But if G has loops, the situation is very diff erent. So the semi-arc automorphism




group of a graph is valuable at least for enumerating maps on surface underlying a graph G

with loops because we need the semi-arc automorphism group, not just the automorphism

group of G in this case.

3.6.6 Considering the local symmetry of a graph, graphs can be seen as the sources of

permutation multigroups. In fact, automorphism of a graph surveys its globally symmetry.

But this can be only applied for that of fields understood by mankind. For the limitation

of recognition, we can only know partially behaviors of World. So a globally symmetry

in one’s eyes is localized symmetry in the real-life World. That is the motivation of

multigroups. Although to determine the automorphism of a graph is very difficult, it is

easily to determine the automorphism multigroups in many cases. Theorems 3.5.3 and

3.5.5 are such typical examples. It should be noted that Theorems 3 .5.4 and 3.5.6 show

that the automorphism multigroups Aut E G and AutclG are new invariants on graphs. So

we can survey localized symmetry of graphs or classify graphs by the action of Aut E G

and AutclG.



CHAPTER 4.

Surface Groups

The surface group is generated by loops on a surface with or without bound-ary. There are two disguises for a surface group in mathematics. One is the

fundamental group in topology and another is the non-Euclidean crystallo-

graphic group, shortly NEC group in geometry. Both of them can be viewed

as an action group on a planar region, enables one to know the structures of

surfaces. Consequently, topics covered in this chapter consist of two parts

also. Sections 4.1.-4.3 are an introduction to topological surfaces, includ-

ing topological spaces, classification theorem of compact surfaces by that

of polygonal presentations under elementary transformations, fundamental

groups, Euler characteristic, · · ·, etc.. These sections 4.4 and 4.5 consist a

general introduction to the theory of Klein surfaces, including the antiana-

lytic functions, planar Klein surfaces, NEC groups and automorphism groups

of Klein surfaces, · · ·, etc.. All of these are the preliminary for finding au-

tomorphism groups of maps on surfaces or Klein surfaces in the following

chapters.



118 Chap.4 Surface Groups

§4.1 SURFACES

4.1.1 Topological Space. Let T be a set. A topology on a set T is a collection C of

subsets of T , called open sets satisfying properties following:

(T1) ∅ ∈ C and T ∈ C ;

(T2) if U 1, U 2 ∈ C , then U 1 ∩ U 2 ∈ C ;

(T3) the union of any collection of open sets is open.

For example, let T = a, b, c and C = ∅, b, a, b, b, c, T . Then C is a topology

on T . Usually, such a topology on a discrete set is called a discrete topology, otherwise,

a continuous topology. A pair (T , C ) consisting of a set T and a topology C on T is

called a topological space and each element in T is called a point of T . Usually, we also

use T to indicate a topological space if its topology is clear in the context. For example,

the Euclidean space Rn for an integer n ≥ 1 is a topological space.

For a point u in a topological space T , its an open neighborhood is an open set U

such that u ∈ U in T and a neighborhood in T is a set containing some of its open

neighborhoods. Similarly, for a subset A of T , a set U is an open neighborhood or

neighborhood of A if U is open itself or a set containing some open neighborhoods of

that set in T . A basis in T is a collection B of subsets of T such that T = ∪ B∈B B and

B1, B2

∈B, x

∈B1

∩ B2 implies that

∃ B3

∈B with x

∈B3

⊂B1

∩ B2 hold.

Let T be a topological space and I = [0, 1] ⊂ R. An arc a in T is defined to be a

continuous mapping a : I → T . We call a(0), a(1) the initial point and end point of a,

respectively. A topological space T is connected if there are no open subspaces A and B

such that S = A ∪ B with A, B ∅ and called arcwise-connected if every two points u, v

in T can be joined by an arc a in T , i.e., a(0) = u and a(1) = v. An arc a : I → T is

a loop based at p if a(0) = a(1) = p ∈ T . A —it degenerated loop e x : I → x ∈ S , i.e.,

mapping each element in I to a point x, usually called a point loop.

A topological space T is called Hausdor ff if each two distinct points have disjoint

neighborhoods and first countable if for each p ∈ T there is a sequence U n of neigh-

borhoods of p such that for any neighborhood U of p, there is an n such that U n ⊂ U . The

topology is called second countable if it has a countable basis.

Let xn be a point sequence in a topological space T . If there is a point x ∈ T such

that for every neighborhood U of u, there is an integer N such that n ≥ N implies xn ∈ U ,

then un is said converges to u or u is a limit point of un in the topological space T .



Sec.4.1 Surfaces 119

4.1.2 Continuous Mapping. For two topological spaces T 1 and T 2 and a point u ∈ T 1,

a mapping ϕ : T 1 → T 2 is called continuous at u if for every neighborhood V of ϕ(u),

there is a neighborhood U of u such that ϕ(U ) ⊂ V . Furthermore, if ϕ is continuous at

each point u in T 1, then ϕ is called a continuous mapping on T 1.For examples, the polynomial function f : R → R determined by f ( x) = an x

n +

an−1 xn−1 + · · · + a1 x + a0 and the linear mapping L : Rn → Rn for an integer n ≥ 1 are

continuous mapping. The following result presents properties of continuous mapping.

Theorem 4.1.1 Let R , S and T be topological spaces. Then

(1) A constant mapping c : R → S is continuous;

(2) The identity mapping Id : R → R is continuous;

(3) If f : R → S

is continuous, then so is the restriction f |U

of f to an open subset

U of R ;

(4) If f : R → S and g : S → T are continuous at x ∈ R and f ( x) ∈ S , then so

is their composition mapping g f : R → T at x.

Proof The results of (1)-(3) is clear by definition. For (4), notice that f and g are

respective continuous at x ∈ R and f ( x) ∈ S . For any open neighborhood W of point

g( f ( x)) ∈ T , g−1(W ) is opened neighborhood of f ( x) in S . Whence, f −1(g−1(W )) is an

opened neighborhood of x in R by definition. Therefore, g( f ) is continuous at x.

A refinement of Theorem 4.1.1(3) enables us to know the following criterion for

continuity of a mapping.

Theorem 4.1.2 Let R and S be topological spaces. Then a mapping f : R → S is

continuous if and only if each point of R has a neighborhood on which f is continuous.

Proof By Theorem 4.1.1(3), we only need to prove the sufficiency of condition. Let

f : R → S be continuous in a neighborhood of each point of R and U ⊂ S . We show

that f −1(U ) is open. In fact, any point x ∈ f −1(U ) has a neighborhood V ( x) on which f

is continuous by assumption. The continuity of f |V ( x) implies that ( f |V ( x))−1

(U ) is open inV ( x). Whence it is also open in R . By definition, we are easily find that

( f |V ( x))−1(U ) = x ∈ R | f ( x) ∈ U = f −1(U )

V ( x),

in f −1(U ) and contains x. Notice that f −1(U ) is a union of all such open sets as x ranges

over f −1(U ). Thus f −1(U ) is open followed by this fact.




For constructing continuous mapping on a union of topological spaces X , the fol-

lowing result is a very useful tool, called the Gluing Lemma.

Theorem 4.1.3 Assume that a topological space X is a finite union of closed subsets:

X =n

i=1

X i. If for some topological space Y , there are continuous maps f i : X i → Y that

agree on overlaps, i.e., f i| X i

X j = f j| X i

X j for all i, j, then there exists a unique continuous

f : X → Y with f | X i = f i for all i.

Proof Obviously, the mapping f defined by

f ( x) = f i( x), x ∈ X i

is the unique well defined mapping from X to Y with restrictions f | X i = f i hold for all i.

So we only need to establish the continuity of f on X . In fact, if U is an open set in Y ,

then

f −1(U ) = X

f −1(U ) = (

ni=1

X i)

f −1(U )

=

ni=1

( X i

f −1(U )) =

ni=1

( X i

f −1

i (U )) =

ni=1

f −1i (U ).

By assumption, each f i is continuous. We know that f −1i (U ) is open in X i. Whence,

f −1(U ) is open in X . Thus f is continuous on X .

Let X be a topological space. A collection C ⊂ P(X ) is called to be a cover of X if

C ∈CC = X .

If each set in C is open, then C is called an opened cover and if |C| is finite, it is called

a finite cover of X . A topological space is compact if there exists a finite cover in its

any opened cover and locally compact if it is Hausdorff with a compact neighborhood for

its each point. As a consequence of Theorem 4.1.3, we can apply the gluing lemma to

ascertain continuous mappings shown in the next.

Corollary 4.1.1 Let Let X and Y be topological spaces and A1, A2, · · · , An be a fi-

nite opened cover of a topological space X . If a mapping f : X → Y is continuous

constrained on each Ai , 1 ≤ i ≤ n, then f is a continuous mapping.

4.1.3 Homeomorphic Space. Let S and T be two topological spaces. They are

homeomorphic if there is a 1 − 1 continuous mapping ϕ : S → T such that the inverse




maping ϕ−1 : T → S is also continuous. Such a mapping ϕ is called a homeomorphic

or topological mapping. A few examples of homeomorphic spaces can be found in the

following.

Example 4.1.1 Each of the following topological space pairs are homeomorphic.

(1) A Euclidean space Rn and an opened unit n-ball Bn = ( x1, x2, · · · , xn) | x21 + x2

2 +

· · · + x2n < 1 ;

(2) A Euclidean plane Rn+1 and a unit sphere S n = ( x1, x2, · · · , xn+1) | x21 + x2

2 + · · ·+

x2n+1 = 1 with one point p = (0, 0, · · · , 0, 1) on it removed.

In fact, define a mapping f from Bn to Rn for (1) by

f ( x1, x2,

· · ·, xn) =

( x1, x2, · · · , xn)

1 − x21 + x2

2 + · · · + x2n

for ∀( x1, x2, · · · , xn) ∈ Bn. Then its inverse is

f −1( x1, x2, · · · , xn) =( x1, x2, · · · , xn)

1 +

x2

1+ x2

2+ · · · + x2

n

for ∀( x1, x2, · · · , xn) ∈ Rn. Clearly, both f and f −1 are continuous. So Bn is homeomorphic

to Rn. For (2), define a mapping f from S n − p to Rn+1 by

f ( x1, x2, · · · , xn+1) = 11 − xn+1

( x1, x2, · · · , xn).

Its inverse f −1 : Rn+1 → S n − p is determined by

f −1( x1, x2, · · · , xn+1) = (t ( x) x1, · · · , t ( x) xn, 1 − t ( x)),

where

t ( x) =2

1 + x21 + x2

2 + · · · + x2n+1

.

Notice that both f and f −

1 are continuous. Thus S n

−p is homeomorphic to Rn+1.

4.1.4 Surface. For an integer n ≥ 1, an n-dimensional topological manifold is a second

countable Hausdorff space such that each point has an open neighborhood homeomorphic

to an open n-dimensional ball Bn = ( x1, x2, · · · , xn)| x21 + x2

2 +· · ·+ x2n < 1 in Rn. We assume

all manifolds is connected considered in this book. A 2-manifold is usually called surface

in literature. Several examples of surfaces are shown in the following.




Example 4.1.1 These 2-manifolds shown in the Fig.4.1.1 are surfaces with boundary.

rectangle cylinderplane torus

Fig.4.1.1

Example 4.1.2 These 2-manifolds shown in the Fig.4.1.2 are surfaces without boundary.

sphere torus

Fig.4.1.2

By definition, we can always distinguish the right-side and left-side when one object

moves along an arc on a surface S . Now let N be a unit normal vector of the surface S .

Consider the result of a normal vector moves along a loop L on surfaces in Fig.4.1.1 and

Fig.4.1.2. We find the direction of N is unchanged as it come back at the original point u.

For example, it moves on the sphere and torus shown in the Fig.4 .1.3 following.

L

u u

sphere torus

O

Fig.4.1.3




Such loops L in Fig.4.1.3 are called orientation-preserving. However, there are also loops

L in surfaces which are not orientation-preserving. In such case, we get the opposite

direction of N as it come back at the original point v. Such a loop is called orientation-

reversing. For example, the process (1)-(3) for getting the famous Mobius strip shown inFig.4.1.4, in where the loop L is an orientation-reversing loop.

A

B

E

A’

B’

E’

(1)

A

B

E

A’

B’

E’

(2)A

E

B(3)

Ã

v

N

L

Fig.4.1.4

A surface S is defined to be orientable if every loop on S is orientation-preserving.

Otherwise, non-orientable if there at least one orientation-reversing loop on S . Whence,

the surfaces in Examples 4.1.1-4.1.2 are orientable and the Mobius strip are non-orientable.

It should be noted that the boundary of a Mobius strip is a closed arc formed by AB′ and

A′ B. Gluing the boundary of a Mobius strip by a 2-dimensional ball B2, we get a non-

orientable surface without boundary, which is usually called crosscap in literature.

4.1.5 Quotient Space. A natural way for constructing surfaces is by the quotient space

from a surface. For introducing such spaces, let X , Y be a topological spaces and

π : X → Y be a surjective and continuous mapping. A subset U ⊂ Y is defined to be

open if and only if π−1(U ) is open in X . Such a topology on Y is called the quotient

topology induced by π, and π is called a quotient mapping. It can be shown easily that thequotient topology is indeed a topology on Y .

Let ∼ be an equivalent relation on X . Denoted by [q] the equivalence class for each

q ∈ X and let X / ∼ be the set of equivalence classes. Now let π : X → X / ∼ be

the natural mapping sending each element q to the equivalence class [q]. Then X / ∼together with the quotient topology determined by π is called the quotient space and π




the projection. For example, the Mobius strip constructed in Fig.4.1.4 is in fact a quotient

space X / ∼, where X is the rectangle AEBA′ E ′ B′, and

π( x) = x′ if | xA′|=

| x′ A′|, x ∈ AB, y ∈ A′ B′, x if x ∈ X \ ( AB ∪ A′ B′).

Applying quotient spaces, we can also construct surfaces without boundary. For ex-

ample, a projective plane is defined to be the quotient space of the 2-sphere by identifying

every pair of diametrically opposite points, i.e., X = ( x1, x2, x3)| x21 + x2

2 + x23 = 1 with

π(− x1, − x2, − x3) = ( x1, x2, x3).

Now let X be a rectangle ABA′ B′ shown in Fig.4.1.5. Then diff erent identification

of points on AB with A′ B′ and AA′ with BB′ yields diff erent surfaces without boundary

shown in Fig.4.1.5,

A A

B

BB

AA

A’

B’

A’

B’

A’

B’

¹

¹

¹

¹

¹

¹

B

sphere S 2 torus T 2

projection plane P2 Klein bottle K 2

a a a a

b

b

a aa a

b

b

b

b

Fig.4.1.5

where the projection π is determined by

π( x) =

x′ if | xA′| = | x′ A′|, x ∈ AB′ B, y ∈ A′ AB,

x if x ∈ X \ ( AB ∪ A′ B′ ∪ AA′ ∪ BB′)



Sec.4.2 Classification Theorem 125

in the sphere,

π( x) =

x′ if | xA′| = | x′ B′|, x ∈ AA′, x′ ∈ BB′,

x” if | xA

|=

| x′ A′

|, x

∈AB, x′

∈A′ B′,

x if x ∈ X \ ( AB ∪ A′ B′ ∪ AA′ ∪ BB′)

in the torus,

π( x) =

x′ if | xB| = | x′ A′|, x ∈ BAA′, x′ ∈ A′ B′ B,

x if x ∈ X \ ( AB ∪ A′ B′ ∪ AA′ ∪ BB′)

in the projection plane and

π( x) =

x′ if | xA′| = | x′ B′|, x ∈ AA′, x′ ∈ BB′,

x” if

| xA

|=

| x” B′

|, x

∈AB, x′

∈A′ B′,

x if x ∈ X \ ( AB ∪ A′ B′ ∪ AA′ ∪ BB′)

in the Klein bottle, respectively.

$4.2 CLASSIFICATION THEOREM

4.2.1 Connected Sum. Let S 1, S 2 be disjoint surfaces. A connected sum of S 1 and S 2,

denoted by S1#S

2is formed by cutting a circular hole on each surface and then gluing the

two surfaces along the boundary of holes.

¹ ¹

¹

¹

¹

¹

A

B

A’

B’

A(A’)

A

B

C

D

A’

B’

C’

D’

C

D

C’

D’D

C

D’

C’

B(B’)

(1) (2) (3)

I

IIII

I

Fig.4.2.1

For example, we show that a Klein bottle constructed in Fig.4.1.5 is in fact the connected

sum of two Mobius strips in Fig.4.2.1, in where, (1) is the Klein bottle in Fig.4.1.5. It

should be noted that the rectangles CDC ′ D′ and DACC ′ B′ D′ are two Mobius strips after

we cut ABA′ B′ along CC ′, DD′ and then glue along AB, A′ B′ in (3).




For a precise definition of connected sum, let D1 ⊂ S 1 and D2 ⊂ S 2 be closed 2-

dimensional discs, i.e., homeomorphic to B2

= ( x1, x2)| x21 + x2

2≤ 1 with boundary ∂ D1,

∂ D2 homeomorphic to S 1 = ( x1, x2)| x21 + x2

2 = 1. Notice that each ∂ Di homeomorphic to

S1

for i = 1, 2. Let h1 : ∂ D1 → S1

and h1 : ∂ D2 → S1

be such homeomorphisms. Thenh−1

2h1 : ∂ D1 → ∂ D2, i.e., there always exists a homeomorphism ∂ D1 → ∂ D2. Chosen

a homoeomorphism h : ∂ D1 → ∂ D2, then S 1#S 2 is defined to be the quotient space

(S 1 ∪ S 2)/h. By definition, S 1#S 2 is clearly a surface and does not dependent on the

choice of D1, D2 and h.

Example 4.2.1 The following connected sums of orientable or non-orientable surfaces

are orientable or non-orientable surfaces.

(1) A connected sum T 2#T 2#· · ·

#T 2

n

of n toruses is orientable. Particularly, T 2#T 2

is called the double torus.

(2) A connected sum P2#P2# · · · #P2 k

of k projection planes is non-orientable. Partic-

ularly, K 2 = P2#P2 as we shown in Fig.4.2.1.

4.2.2 Polygonal Presentation. A triangulation of a surface S consisting of a finite

family of closed subsets T 1, T 2, · · · , T n that covers S with T i ∩ T j = ∅, a vertex v or an

entire edge e in common, and a family of homeomorphisms φi : T ′i

→ T i, where each T ′i

is a triangle in the plane R

2

, i.e., a compact subset bounded by 3 distinct straight lines.The images of vertices and edges of the triangle T ′

i under φi are called also the vertices

and edges, respectively. For example, a triangulation of the Mobius strip can be found in

Fig.4.2.2.

v1 v2 v3 v4 v5

u1 u2 u3 u4 u5

Fig.4.2.2

In fact, there are many non-isomorphic triangulation for a surface, which is the central

problem of enumerative theory of maps (See [Liu2]-[Liu4] for details). T.Rado proved

the following result in 1925.




Theorem 4.2.1(Rado) Any compact surface S admits a triangulation.

The proof of this theorem is not difficult but very tedious. We will not present it

here. The reader can refers references, such as those of [AhS1] and [Lee1] for details.

The following result is fundamental for classifying surfaces without boundary.

Theorem 4.2.2 Let S be a compact surface with a triangulation T . Then S is homeo-

morphic to a quotient surface by identifying edge pairs of triangles in T .

Proof Let T = T i; 1 ≤ i ≤ n be a triangulation of S . Our proof is divided into two

assertions following:

(A1) Let v be a vertex of T . Then there is an arrangement of triangles with v as a

vertex in cyclic order T v1

, T v2

,

· · ·, T v

ρ(v)such that T i and T i+1 have an edge in common for

integers 1 ≤ i ≤ ρ(v) (mod ρ(v)).

Define an equivalence on two triangles T vi

, T v j

by that of T vi

and T v j

have exactly an

edge in common in T . It is clear that this relation is indeed an equivalent relation on T .Denote by [T ] all such equivalent classes in T . Then if |[T ]| = 1, we get the assertion

(A1). Otherwise, |[T ]| ≥ 2, we can choose [T vs ], [T vl

] ∈ [T ] such that [T vs ] ∩ [T vl

] = v in

T . Whence, there is a neighborhood W v of v small enough such that W v−v is disconnected.

But by the definition of surface, there is a neighborhood W v of v homeomorphic to an open

sphere B2 in S . Consequently, W v

−v is connected for any neighborhood W v of v small

enough, a contradiction.

(A2) Each edge is an edge of exactly two triangles.

First, each edge is an edge of two triangles at least in T , i.e., there are no vertices

x on an edge of T i for an integer, i, 1 ≤ i ≤ n with a neighborhood W x homeomorphic

to an open ball B2. Otherwise, a loop L encircled x in T i − W x can not be continuously

contracted to the point in T i. But it is clear that any loop in T i − W x for neighborhoods W x

of x small enough can be continuously contracted to a point in T i − W x for any point x on

an edge of T x, a contradiction.

Second, each edge is exactly an edge of two triangles. Notice that we can continu-

ously subdivide a triangulation such that triangles T with a common edge e are contained

in an ǫ -neighborhood of a point in T . Not loss of generality, we assume T is such a trian-

gulation of S . By applying Jordan curve theorem, i.e., the moving of any closed curve C on

S 2 reminds two connected components W 1, W 2 with W 1∩W 2 = C , we know that each edge




is exactly an edge of two triangles in T . In fact, let ee11e21, ee12e22, · · · , ees1es1 be trian-

gles contained in an ǫ -neighborhood W with a common edge e, where e, e1i, e2i, 1 ≤ i ≤ s

are edges of these triangles. Then W − ee11e21 has two connected components by Jordan

curve theorem. One of them is the interior of triangle ee11e21 and another is W −T e, whereT e is the triangle with boundary ee11e21. So there must be s = 2.

Combining assertions (A1)-(A2), we consequently get the result.

According to Theorem 4.2.2, we know that a compact surface can be presented by

identifying edges of triangles, where each edge is exactly an edge of two triangles. Gen-

erally, let A be a set. A word is defined to be an ordered k -tuple of elements a ∈ A with

the form a or a−1. A polygonal presentation, denoted by

W = A | W 1, W 2, · · · , W k

isa finite set A together with finitely many words W 1, W 2, · · · , W k in A such that each ele-

ment of A appears in at least one words. A polygonal presentation A |W 1, W 2, · · · , W k is

called a surface presentation if each element a ∈ A occurs exactly twice in W 1, W 2, · · · , W k

with the form a or a−1. We call elements a ∈ A to be edges, W i, 1 ≤ i ≤ k to be faces

of S and vertices appeared in each face vertices if each words is represented by a poly-

gon on the plane R2. It can be known that a surface is orientable if and only if the two

occurrences of each element a ∈ A are with diff erent power, otherwise, non-orientable.

For example, let S be the torus T 2 with short side a and length side b in Fig.4.1.5.

Then we get its polygonal presentation T 2 =

a, b|aba−1b−1

. Generally, Theorem 4.2.2

enables one knowing that the existence of polygonal presentation for compact surfaces S ,

at least by triangles, i.e., each words W is length of 3 in A .

4.2.3 Elementary Equivalence. Let A be a set of English alphabets, the minuscules

a, b, c, · · · ∈ A but the Greek alphabets α,β,γ, · · · A , S = A |W 1, W 2, · · · , W k be a

surface presentation and let the capital letters A, B,· · ·

be sections of successive elements

in order and A−1, B−1, · · · in reserving order in words W . For two words W 1, W 2 in S, the

notation W 1W 2 denotes the word formed by concatenating W 1 with W 2 in order. We adopt

the convention that (a−1)−1 = a in this book.

Define operations El.1–El.6, called elementary transformations on S following:

El.1(Relabeling): Changing all occurrences of a by α A , interchanging all oc-




currences of two elements a and b, or interchanging all occurrences a and a−1 , i.e.,

A |aAbB, W 2, · · · , W k ↔ A |bAaB, W 2, · · · , W k ,

A |aAa−1 B, W

2,· · ·

, W k ↔ A |

a−1 Aa, W 2

,· · ·

, W k or

A |aA, a−1 B, · · · , W k

↔ A |a−1 A, aB, · · · , W k

.

El.2(Subdividing or Consolidating) Replacing every occurrence of a by αβ and a−1

by β−1α−1 , or vice versa, i.e.,A |aAa−1 B, W 2, · · · , W k

↔

A |αβ A β−1α−1 B, W 2, · · · , W k

A |aA, a−1 B, · · · , W k

↔

A |αβ A, β−1α−1 B, · · · , W k

.

El.3(Reflecting) Reversing the order of a word W = a1a2

· · ·am , i.e.,

A |a1, a2 · · · am, W 2, · · · , W k ↔

A |a−1m · · · a−1

2 a−11 , W 2, · · · , W k

.

El.4(Rotating) Changing the order of a word W = a1a2 · · · am by rotating, i.e.,

A |a1, a2 · · · am, W 2, · · · , W k ↔ A |ama1 · · · am−1, W 2, · · · , W k .

El.5(Cutting or Pasting) If the length of W 1, W 2 are both not less than 2 , then

A |W 1W 2, · · · , W k ↔

A |W 1γ, γ −1W 2, · · · , W k

.

El.6(Folding or Unfolding) If the length of W 1 is at least 3 , thenA |W 1δδ−1, W 2, · · · , W k

↔ A |W 1, W 2, · · · , W k .

Let S1 and S2 be two surface presentations. If S1 can be conversed to that of S2 by

a series of elementary transformations π1, π2, · · · , πm in El.1 − − El.6, we say S1 and S2

to be elementary equivalent and denote by S1 ∼ El S2. It is obvious that the elementary

equivalence is indeed an equivalent relation on surface presentations. The following result

is fundamental for applying surface presentations to that of classifying compact surfaces.

Theorem 4.2.3 Let S 1 and S 2 be compact surfaces with respective presentations S1 , S2.

If S1 ∼ El S2 , then S 1 is homeomorphic to S 2.

Proof By the definition of elementary transformation, it is clear that each pairs of

cutting and pasting, folding and unfolding, subdividing and consolidating are inverses of

each other. Whence, we are only need to prove our result for one of such pairs.




Cutting. Let P1 and P2 be convex polygons labeled by W 1γ and γ −1W 2, respectively

and P be a convex polygon labeled by W 1W 2. Not loss of generality, we assume these are

the only words in their respective presentations. Let π : P1 ∪ P2/ ∼→ S 1 and π′ : P/ ∼→S 2 be the quotient mappings. The line segment going from the terminal vertex of W 1 inP to its initial vertex lies in P by convexity, labeled this line segment by γ . Such as those

shown in Fig.4.2.3 following.

W 1 W 1

W 2 W 2γ

Ã

Ã

γ γ pasting

cutting

Fig.4.2.3

Applying the gluing lemma, there is a continuous mapping f : P1 ∪ P2 → P that takes

each edge of P1 or P2 to the edge in P with a corresponding label, and whose restriction

to P1 or P2 is a homeomorphism, i.e., f is a quotient mapping. Because f identifying two

edges labeled by γ and γ −1 but nothing else, the quotient mapping π f and π′ makes the

same identifications. So their quotient spaces are homeomorphic.

a

b

c

e

e a

b

ce©

Í

¹

Ê

«

¹

Ã

folding

unfolding

Fig.4.2.4

If k ≥ 3, extending f by declaring it to be the identity on the respective polygons and

processed as above, we also get the result.

Folding. Similarly, we can ignore the additional words W 2, · · · , W k . If the length of

W 1 is 2, subdivide it and then perform the folding transformation and then consolidate.

So we can assume the length of W 1 is not less than 3. First, let W 1 = abc and P, P′ be

convex polygons with edge labels abcee−1 and abc, respectively. Let π : P → S 1 and

π′ : P′ → S 2 be the quotient mappings. Now adding edges in P, P′, turns them into




polyhedra, such as those shown in Fig.4.2.4. There is a continuous mapping f : P → P′

that takes each edge of P to that the edge of P′ with the same label. Then π′ f and π are

quotient mappings that make the same identifications.

If the length≥ 4 of W 1, we can write W 1 = Abc for some section A of length at least2. Cutting along a we obtain

A , b, c, e| Abcee−1

∼ El

A , a, b, c, e| Aa−1, abcee−1

and processed as before to get the result.

Subdividing. Similarly, let P1, P2 be distinct polygons with sections a or a−1 and

P′1

, P′2 with sections replacing a by αβ and a−1 by β−1α−1 in P1 and P2. Such as those

shown in Fig.4.2.5.

subdividing

consolidating

a

α

β

a

Ã

Fig.4.2.5

Certainly, there is a continuous mapping f : P1 ∪ P2 → P′1∪ P′

2 that takes each edge of

P1, P2 to that the edge of P′1

, P′2 with the same label, and the edge with label a to the edge

with label αβ in P′1∪ P′

2. Then π′ f : P1 ∪ P2/ ∼→ S 1 and π : P′1∪ P′

2/ ∼→ S 2 are

quotient mappings that make the same identifications.

If a or a−1 appears twice in a polygon P, the proof is similar. Thus S 1 is homeomor-

phic to S 2 in each case.

4.2.4 Classification Theorem. Let S be a compact surface with a presentation S =

A |W

1, W

2,· · ·

, W k

and let A, B,· · ·

be sections of successive elements in a word W inS

.

Theorems 4.2.1–4.2.3 enables one to classify compact surfaces as follows.

Theorem 4.2.4 Any connected compact surface S is either homeomorphic to a sphere,

or to a connected sum of tori, or to a connected sum of projective planes, i.e., its sur-

face presentation S is elementary equivalent to one of the standard surface presentations

following:




(1) The sphere S 2 =

a|aa−1

;

(2) The connected sum of p tori

T 2#T 2#· · ·

#T 2 p

= ai, bi, 1≤

i≤

p|

p

i=1

aibia−1i b−1

i ;

(3) The connected sum of q projective planes

P2#P2 · · · #P2 q

=

ai, 1 ≤ i ≤ q |

qi=1

ai

.

Proof Let S = A |W 1, W 2, · · · , W k . For establishing this theorem, we first prove

several claims on elementary equivalent presentations of surfaces following.

Claim 1. There is a word W in A such that

S = A | W 1, W 2, · · · , W k ∼ El A | W .

If k ≥ 2, we can concatenate W 1, W 2, · · · , W k by elementary transformations El.1 − El.6. In fact, by definition, there is an element a only appears once in W 1. Thus W 1 = Aa

and a does not appears in A. Not loss of generality, let a or a−1 appears in W 2, i.e.,

W 2 = Ba or W 2 = a−1 B. Applying El.1 − El.6, we know that

S = A | Aa, Ba, W 3, · · · , W k ∼ El

A | Aa, a−1 B−1, W 3, · · · , W k ∼ El

A | AB−1, W 3, · · · , W k .

S =

A | Aa, a−1 B, W 3, · · · , W k

∼ El A | AB, W 3, · · · , W k .

Furthermore, by induction on k we know that S is elementary equivalent to a surface just

with one word W if k ≥ 2. Thus

S = A | W 1, W 2, · · · , W k ∼ El A | W .

Claim 2.

A | AaBbCa−1 Db−1 E

∼ El

A | ADCBEaba−1b−1

.

In fact, by El.1 − El.6, we know that

A | AaBbCa−1 Db−1 E ∼ El A ∪ δ | Db−1 EAaδ, δ−1 BbCa−1

∼ El

A ∪ δ \ b | EAaδ DCa−1δ−1 B

∼ El

A ∪ δ | Aaδb, b−1 DCa−1δ−1 BE

∼ El

A | bAaBEb−1 DCa−1

∼ El

A ∪ δ | AaBE δ, δ−1b−1 DCa−1b

∼ El

A ∪ δ \ a | BE δ Abδ−1b−1 DC

∼ El

A ∪ δ | Aba, a−1δ−1b−1 DCBE δ

∼ El

A ∪ δ \ b | ADCBE δaδ−1a−1

∼ El


.






Claim 2 we know that S ∼ El


. Notice that elements in ADCBE

also satisfy the condition of Case 1. So we can applying Claim 2 repeatedly and finally

get that

S ∼ El A | p

i=1

aibiaib−1i

for an integer p ≥ 1.

Case 2. There are elements a ∈ A such that W = AaBaC.

In this case, by Claim 3 we know that S ∼ El

A | AB−1Caa

. Applying Claim 3 to

AB−1C repeatedly, we finally get that

S ∼ El A

|H

s

i=1

aiaifor an integer s ≥ 1 such that there are no elements b ∈ H such that H = DbCbE . Thus

each element x ∈ A \ ai; 1 ≤ i ≤ s appears x at one time and x−1 at another. Similar to

the discussion of Case 1, we know that

S ∼ El

A | H

si=1

aiai

∼ El

A |

si=1

aiai

t i=1

x j y j x−1 j y−1

for some integers s, t by applying Claim 2. Applying Claim 4 also, we finally get that

S ∼ El

A | H

si=1

aiai

∼ El

A |

qi=1

aiai

,

for an integer q = s + 2t . This completes the proof.

Notice that each step in the proof of Theorem 4.2.4 does not change the orientability

of a surface S with a presentation S. We get the following conclusion.

Corollary 4.2.1 A surface S is orientable if and only if it is elementary equivalent to the

sphere S 2 or the connected sum T 2#T 2#· · ·

#T 2 p

of p tori.

4.2.5 Euler Characteristic. Let S = A | W 1, W 2, · · · , W k be a surface presentation

and π : A | W 1, W 2, · · · , W k → S a projection by identifying a with a−1 for ∀a ∈ A .

The Euler characteristic of S is defined by

χ(S) = |V (S)| − | E (S)| + |F (S)|,




where V (S), E (S) and F (S) are respective the set of vertex set, edge set and face set of the

surface S. We are easily knowing that | E (S)| = |A |, |F (F )| = k and |V (S)| the number of

orbits of vertices in polygons W 1, W 2, · · · , W k under π. The Euler characteristic of a sur-

face is topological invariant. Furthermore, it is unchange by elementary transformations.

Theorem 4.2.5 If S1 ∼ El S2 , then χ(S1) = χ(S2) , i.e., the Euler characteristic is an

invariant under elementary transformations.

Proof Let A | W 1, W 2, · · · , W k be a presentation of a surface S. We only need

to prove each elementary El.1 − El.6 on S does not change the value χ(S). Notice the

elementary transformations El.1(Relabeling), El.3(Reflecting) and El.4(Rotating) leave

the numbers of vertices, edges and faces unchanged. Consequently, χ(S) is invariant

under El.1, El.3−

El.4. We only need to check the result for elementary transforma-

tions El.2(Subdividing or Consolidating), El.5(Cutting or Pasting) and El.6(Folding or

Unfolding). In fact, El.2(Subdividing or Consolidating) increase or decrease both the

number of edges and the number of vertices by 1, leaves the number of faces unchanged,

El.5(Cutting or Pasting) increases or decreases both the number of edges and the number

of faces by 1, leaves the number of vertices unchanged and El.6(Folding or Unfolding)

increases or decreases the number of edges and the number of vertices, leaves the number

of faces unchanged. Whence, χ(S) is invariant under these elementary transformations

El.1

− El.6. This completes the proof.

Applying Theorems 4.2.4 and 4.2.5, we get the Euler characteristic of connected

compact surfaces following.

Theorem 4.2.6 Let S be a connected compact surface with a presentation S. Then

χ(S ) =

2, if S ∼ El S 2,

2 − 2 p, if S ∼ El T 2#T 2# · · · #T 2 p

,

2 − q, if S ∼ El P2#P2# · · · #P2

q

.

Proof Notice that the numbers of vertices, edges and faces of a surface S are re-

spective |V (S)| = 2, | E (S)| = 1, |F (S)| = 1 if S =

a|aa−1

(See Fig.4.1.5 for de-

tails), |V (S)| = 1, | E (S)| = 2 p, |F (S)| = 1 if S =

ai, bi, 1 ≤ i ≤ p |

pi=1

aibia−1i

b−1i

and

|V (S)| = 1, | E (S)| = q, |F (S)| = 1 if S =

ai, 1 ≤ i ≤ q |

qi=1

ai

. By definition, we know




that

χ(S ) =

2, if S ∼ El S 2,

2 − 2 p, if S ∼ El T 2#T 2# · · · #T 2

p

,

2 − q, if S ∼ El P2#P2# · · · #P2 q

by Theorem 4.2.5. Applying Theorems 4.2.4, the conclusion is followed.

The numbers p and q is usually defined to be the genus of the surface S , denoted

by g(S ). Theorem 4.2.6 implies that g(S ) = 0, p or q if S is elementary equivalent to the

sphere, the connected sum of p tori or the connected sum of q projective plane.

$4.3 FUNDAMENTAL GROUPS

4.3.1 Homotopic Mapping. Let T 1, T 2 be two topological spaces and let ϕ1, ϕ2 : T 1 →T 2 be two continuous mappings. If there exists a continuous mapping H : T 1 × I → T 2

such that

H ( x, 0) = ϕ1( x) and H ( x, 1) = ϕ2( x)

for ∀ x ∈ T 1, then ϕ1 and ϕ2 are called homotopic, denoted by ϕ1 ≃ ϕ2. Furthermore, if

there is a subset A ⊂ T such that

H (a, t ) = ϕ1(a) = ϕ2(a), a ∈ A, t ∈ I ,

then ϕ1 and ϕ2 are called homotopic relative to A. Clearly, ϕ1 is homotopic to ϕ2 if A = ∅.

Theorem 4.3.1 For two topological spaces T , J , the homotopic ≃ on the set of all

continuous mappings from T to J is an equivalent relation, i.e, all homotopic mappings

to a mapping f is an equivalent class, denoted by [ f ].

Proof Let f , g, h be continuous mappings from T to J , f ≃ g and g ≃ h with

homotopic mappings H 1 and H 2. Then we know that

(1) f ≃ f if choose H : I × I → T by H (t , s) = f (t ) for ∀s ∈ I .

(2) g ≃ f if choose H (t , s) = H 1(t , 1 − s) for ∀s, t ∈ I which is obviously continuous.

(3) Define H (t , s) = H 2 H 1(t , s) for ∀s, t ∈ I by

H (t , s) = H 2 H 1(t , s) =

H 1( x, 2t ), if 0 ≤ t ≤ 12

,

H 2( x, 2t − 1), if 12

≤ t ≤ 1.



Sec.4.3 Fundamental Groups 137

Notice that H 1( x, 2t ) = H 1( x, 1) = g( x) = H 2( x, 2t − 1) if t = 12

. Applying Theorem 4.1.3,

we know the continuousness of H 1 H 2. Whence, f ≃ h.

Theorem 4.3.2 If f 1, f 2 : T

→J and g1, g2 : J

→L are continuous mappings with

f 1 ≃ f 2 and g1 ≃ g2 , then f 1 g1 ≃ f 2 g2.

Proof Assume F : f 1 ≃ f 2 and G : g1 ≃ g2 are homotopies. Define a new homotopy

H : T × I → L by H ( x, t ) = G(F ( x, t ), t ). Then H ( x, 0) = G( f 1( x), 0) = g1( f 1( x)) for

t = 0 and H ( x, 1) = G( f 2( x), 1) = g2( f 2( x)) for t = 1. Thus H is a homopoty from g1 f 1

to g2 f 2.

We present two examples for homotopies of topological spaces.

Example 4.3.1 Let f , g : R → R2 determined by

f ( x) = ( x, x2), g( x) = ( x, x)

and H ( x, t ) = ( x, x2 − tx2 + tx). Then H : R × I → R is continuous with H ( x, 0) = f ( x)

and H ( x, 1) = g( x). Whence, H : f ≃ g.

Example 4.3.2 Let f , g : T → R2 be continuous mappings from a topological space T

to R2. Define a mapping H : T × I → T by

H ( x, t ) = (1 − t ) f ( x) + tg( x), x ∈ T .

Clearly, H is continuous with H ( x, 0) = f ( x) and H ( x, 1) = g( x). Therefore, H : f ≃ g.

Such a homotopy H is called a straight-line homotopy between f and g.

4.3.2 Fundamental Group. Particularly, let a, b : I → T be two arcs with a(0) = b(0)

and a(1) = b(1) in a topological space T . In this case, a ≃ b implies that there exists a

continuous mapping

H : I × I → S

such that H (t , 0) = a(t ), H (t , 1) = b(t ) for

∀t

∈I by definition.

Now let a and b be two arcs in a topological space T with a(1) = b(0). A product

arc a · b of a with b is defined by

a · b(t ) =

a(2t ), if 0 ≤ t ≤ 12

,

b(2t − 1), if 12

≤ t ≤ 1

and an inverse mapping of a by a = a(1 − t ).




Notice that a ·b : I → T and a : I → T are continuous by Corollary 4.1.1. Whence,

they are indeed arcs by definition, called the product arc of a with b and the inverse arc

of a. Sometimes it is needed to distinguish the orientation of an arc. We say the arc a

orientation-preserving and its inverse a orientation-reversing.Let a, b be arcs in a topological space T . Properties on product of arcs following

are hold obviously by definition.

(P1) a = a;

(P2) b · a = a · b providing ab existing;

(P3) e x = e x, where x = e(0) = e(1).

Theorem 4.3.3 Let a, b, c and d be arcs in a topological space S . Then

(1) a ≃ b if a ≃ b;

(2) a · b ≃ c · d if a ≃ b, c ≃ d with a · c an arc.

proof Let H 1 be a homotopic mapping from a to b. Define a continuous mapping

H ′ : I × I → S by H ′(t , s) = H 1(1 − t , s) for ∀t , s ∈ I . Then we find that H ′(t , 0) = a(t )

and H ′(t , 1) = b(t ). Whence, we get that a ≃ b, i.e., the assertion (1).

For (2), let H 2 be a homotopic mapping from c to d . Define a mapping H : I × I → S

by

H (t , s) = H 1(2t , s), if 0

≤t

≤12

,

H 2(2t − 1, s), if 12

≤ t ≤ 1.

Notice that a(1) = c(0) and H 1(1, s) = a(1) = c(0) = H 2(0, s). Applying Corollary 4.1.1,

we know that H is continuous. Therefore, a · b ≃ c · d .

For a topological space T , x0 ∈ T , let π1(T , x0) be a set consisting of equivalent

classes of loops based at x0. Define an operation in π1(T , x0) by

[a] [b] = [a · b] and [a]−1 = [a−1].

Then we know that π1(T , x0) is a group shown in the following result.

Theorem 4.3.4 π1(T , x0) is a group.

Proof We check each condition of a group for π1(T , x0). First, it is closed under

the operation since [a] [b] = [a · b] is an equivalent class of loop a · b based at x0 for

∀[a], [b] ∈ π1(T , x0).




Now let a, b, c : I → T be three loops based at x0. By definition we know that

(a

·b)

·c(t ) =

a(4t ), if 0 ≤ t ≤ 14

,

b(4t

−1), if 1

4 ≤t

≤1

2

,

c(2t − 1), if 12

≤ t ≤ 1.

and

a · (b · c)(t ) =

a(2t ), if 0 ≤ t ≤ 1

2,

b(4t − 2), if 12

≤ t ≤ 34

,

c(4t − 3), if 34

≤ t ≤ 1.

Define a function H : I × I → T by

H (t , s) =

a(4t

1 + s), if 0

≤t ≤

s + 1

4,

b(4t − 1 − s), if s + 1

4≤ t ≤ s + 2

4,

c(1 − 4(1 − t )

2 − s), if

s + 2

4≤ t ≤ 1.

Then H is continuous by applying Corollary 4.1.1, H (t , 0) = ((a · b) · c)(t ) and H (t , 1) =

(a · (b · c))(t ). Thereafter, we know that ([a] [b]) [c] = [a] ([b] [c]).

Now let e x0: I → x0 ∈ T be the point loop at x0. Then it is easily to check that

a

·a

≃e x0

, a

·a

≃e x0

and

e x0· a ≃ a, a · e x0

≃ a.

We conclude that π1(T , x0) is a group with a unit [e x0] and an inverse element [a−1]

for any [a] ∈ π1(S , x0) by definition.

Let T be a topological space, x0, x1 ∈ T and £ an arc from x0 to x1. For ∀[a] ∈π1(T , x0), we know that £ [a] £−1 ∈ π1(T , x1) (see Fig.4.31.1 below). Whence, the

mapping £# = £

[a]

£−1 : π1(T , x0)→

π1(T , x1).

¶

x0

x1

£

[a]

Fig.4.3.1





Theorem 4.3.5 Let T be a topological space. If x0, x1 ∈ T and £ is an arc from x0 to x1

in T , then π1(T , x0)

≃π1(T , x1).

Proof We have known that £# : π1(T , x0) → π1(T , x1). For [a], [b] ∈ π1(T , x0),

[a] [b], we find that

£#([a]) = £ [a] £−1 £ [b] £−1 = £#([b]),

i.e., £# is a 1 − 1 mapping. Choose [c] ∈ π1(T , x0). Then

£#([a]) £#([c]) = £ [a] £−1 £ [b] £−1 = £ [a] e x1 [a] £−1

= £ [a] [b] £−1 = £#([a] [b]).

Therefore, £# is a homomorphism.

Similarly, £−1# = £−1 [a] £ is also a homomorphism from π1(T , x1) to π1(T , x0)

and £−1#

£# = [e x1], £# £−1

# = [e x0] are the identity mappings between π1(T , x0) and

π1(T , x1).Hence, £# is an isomorphism form π1(T , x0) to π1(T , x1).

Theorem 4.3.5 implies the fundamental group of a arcwise-connected space T is

independent on the choice of base point x0. Whence, we can denote the fundamental

group of T by π1(T ). If π1(T ) = [e x0], then T is called to be a simply connected

space. For example, the Euclidean space Rn, n-ball Bn are simply connected spaces for

n ≥ 2. We determine the fundamental groups of graphs embedded in topological spaces

in the followiing.

Theorem 4.3.6 Let G be an embedded graph on a topological space S and T a spanning

tree in G. Then π1(G) = T + e | e ∈ E (G \ T ) .

Proof We prove this assertion by induction on the number of n = | E (T )|. If n = 0,

G is a bouquet, then each edge e is a loop itself. A closed walk on G is a combination of

edges e in E (G), i.e., π1(G) = e | e ∈ E (G) in this case.

Assume the assertion is true for n = k , i.e., π1(G) = T + e | e ∈ E (G \ T ) . Con-

sider the case of n = k + 1. For any edgee ∈ E (T ), we consider the embedded graph

G/e, which means continuously to contract e to a point v in S . A closed walk on G

passes or not throughe in G is homotopic to a walk passes or not through v in G/e for

κ (T ) = 1. Therefore, we conclude that π1(G) = T + e | e ∈ E (G \ T ) by the induction

assumption.




4.3.3 Seifert-Van Kampen Theorem. For a subset A of B, an inclusion mapping i :

A → B is defined by i(a) = a for ∀a ∈ A. A subset A of a topological space X is called a

deformation retract of X if there exists a continuous mapping r : X → A and a homotopy

f : X × I → X such that

f ( x, 0) = x, f ( x, 1) = r ( x), ∀ x ∈ X and f (a, t ) = a, ∀a ∈ A and t ∈ I .

we have the following result.

Theorem 4.3.7 If A is a deformation retract of X, then the inclusion mapping i : A → X

induces an isomorphism of π1( A, a) onto π1( X , a) for any a ∈ A.

Proof Let i∗ : π1( A, a) → π1( X , a) and r ∗ : π1( X , a) → π1( A, a) be induced homo-

morphisms by i and r . We conclude that r ∗i∗ is the identity mapping of π1( A, a). Noticethat ir is homotopic to the identity mapping X → X relative to a. We know that i∗r ∗ is

the identity mapping of π1( X , a). Thus i∗ : π1( A, a) → π1( X , a) is an isomorphism.

Generally, to determine the fundamental group π1(T ) of a topological space T is not

easy, particularly for finding its presentation. For this objective, a useful tool is the Seifert-

Van Kampen theorem. Its modern form is presented by homomorphisms following.

Theorem 4.3.8(Seifert and Van-Kampen) Let X = U ∪ V with U , V open subsets and let

X , U , V, U

∩V be non-empty arcwise-connected with x0

∈U

∩V and H a group. If there

are homomorphisms

φ1 : π1(U , x0) → H and φ2 : π1(V , x0) → H

and

π1(U ∩ V , x0)

π1(U , x0)

π1( X , x0)

π1(V , x0)

H

¹

¹

¹ ¹

i1

i2

φ1

φ2

j1

j2

Φ

with φ1 · i1 = φ2 · i2 , where i1 : π1(U ∩ V , x0) → π1(U , x0) , i2 : π1(U ∩ V , x0) → π1(V , x0) ,

j1 : π1(U , x0) → π1( X , x0) and j2 : π1(V , x0) → π1( X , x0) are homomorphisms induced by




inclusion mappings, then there exists a unique homomorphism Φ : π1( X , x0) → H such

that Φ · j1 = φ1 and Φ · j2 = φ2.

The classical form of the Seifert-Van Kampen theorem is by the following.

Theorem 4.3.9(Seifert and Van-Kampen theorem, Classical Version) Let X = U ∪ V

with U , V open subsets and let X , U , V, U ∩ V be non-empty arcwise-connected with

x0 ∈ U ∩ V, inclusion mappings i1, j1, i2, j2 as the same in Theorem 4.3.7. If

j : π1(U , x0) ∗ π1(V , x0) → π1( X , x0)

is an extension homomorphism of j1 and j2 , then j is an epimorphism with kernel Ker j

generated by i−11 (g)i2(g), g ∈ π1(U ∩ V , x0) , i.e.,

π1( X , x0) ≃π1(U , x0)

∗π1(V , x0)i−1

1 (g) · i2(g)| g ∈ π1(U ∩ V , x0) ,

where [ A] denotes the minimal normal subgroup of a group G included A ⊂ G .

A complete proof of the Seifert-Van Kampen theorem can be found in references,

such as those of [Lee1] [Mas1] or [Mun1]. By this result, we immediately get the follow-

ing conclusions.

Corollary 4.3.1 Let X 1, X 2 be two open sets of a topological space X with X = X 1 ∪ X 2 ,

X 2 simply connected and X , X 1 and X 0 = X 1 ∩ X 2 non-empty arcwise-connected, then for

∀ x0 ∈ X 0 ,

π1( X , x0) ≃ π1( X 1, x0)

[ (i1)π([a])|[a] ∈ π1( X 0, x0) ].

Corollary 4.3.2 Let X 1, X 2 be two open sets of a topological space X with X = X 1 ∪ X 2.

If there X , X 1, X 2 are non-empty arcwise-connected and X 0 = X 1 ∩ X 2 simply connected,

then for ∀ x0 ∈ X 0 ,

π1( X , x0) ≃ π1( X 1, x0)π1( X 2, x0).

Corollary 4.3.2 can be applied to find the fundamental group of an embedded graph,

particularly, a bouquet Bn =

n

i=1

Li consisting of n loops Li, 1 ≤ i ≤ n again following,

which is the same as in Theorem 4.3.6.

Let x0 be the common point in Bn. For n = 2, let U = B2 − x1, V = B2 − x2, where

x1 ∈ L1 and x2 ∈ L2. Then U ∩ V is simply connected. Applying Corollary 3.1.2, we get

that

π1( B2, x0) ≃ π1(U , x0)π1(V , x0) ≃ L1 L2 = L1, L2 .




Generally, let xi ∈ Li, W i = Li − xi for 1 ≤ i ≤ n and

U = L1

W 2

· · ·

W n and V = W 1

L2

· · ·

Ln.

Then U V = S 1.n, an arcwise-connected star. Whence,

π1( Bn, O) = π1(U , O) ∗ π1(V , O) ≃ L1 ∗ π1( Bn−1, O).

By induction induction, we finally find the fundamental group

π1( Bn, O) = Li, 1 ≤ i ≤ n .

4.3.4 Fundamental Group of Surface. Applying the Seifert-Van Kampen theorem and

the classification theorem of connected compact surfaces, we can easily get the funda-

mental groups following, usually called the surface groups in literature.

Theorem 4.3.10 The fundamental groups π1(S ) of compact surfaces S are respective

π1(S ) =

1 , the trivial group if S ∼ El S 2;a1, b1, · · · , a p, b p |

pi=1

aibia−1i b−1

i = 1

if S ∼ El T 2#T 2# · · · #T 2

p

;c1, c2, · · · , cq |

qi=1

c2i = 1

if S ∼ El P2#P2# · · · #P2

q

,

Proof If S ∼ El S 2, then it is clearly that π1(S ) is trivial. Whence, we consider S is

elementary equivalent to the connected sum of p tori or q projective planes following.

Case 1. S ∼ El T 2#T 2# · · · #T 2 p

.

Let S =

a1, b1, · · · , a p, b p |

pi=1

aibia−1i

b−1i

be the surface representation of S . By

Theorem 4.2.2, we can represent S by a 4 p-gon on the plane with sides identified in pairs

such as those shown in Fig.4.3.2(a). By the identification, these edges a1, b1, a2, b2, · · · , a p, b p

become circuits, and any two of them intersect only in the base point x0. Now let

U = S \ y, the complement of the center y and let V be the image of the interior of

the 4 p-gon under the identification. Then U , V both are arewise-connected. Furthermore,

the union of circuits a1, b1, a2, b2, · · · , a p, b p is a deformation retract of U , and V is simply

connected. Therefore,

π1(V , x1) = 1 |∅ , π1(U , x0) =

α1, β1, α2, β2, · · · , α p, β p | ∅

,








Particularly, let f : A → C be determined by f : z = x + iy → f ( z) = u( x, y) − iv( x, y).

Then we get the fundamental equalities following:

∂ f

∂ z = ∂ f

∂ z , ∂ f

∂ z = ∂ f

∂ z . (4 − 1)

Let C+ = z | Im z ≥ 0 . A mapping f : A −→ C (or C+) is called to be analytic

on A if ∂ f

∂ z= 0 (Cauchy-Riemann equation) and antianlytic on A if

∂ f

∂ z= 0. A mapping

f : A → C (or C+) is dianalytic if its restriction to every connected component of A

is analytic or antianalytic. The following properties of dianalytic mappings is clearly by

formulae (4-1) and definition.

(P1) A mapping f : A → C (or C+) is analytic if and only if f is antianalytic;

(P2) If a mapping f : A→ C

(or C

+) is both analytic and antianalytic, then f is

constant;

(P3) If f : A → B ⊂ C (or C+) and g : B → C (or C+) are both analytic or

antianalytic, then the composition g f : A → C (or C+) is analytic. Otherwise, g f is

antianalytic.

Example 4.4.1 Let a, b, c, d ∈ R, c 0 and A = C \ −d /c. Clearly, the mapping

f : A → C determined by f ( z) =az + b

cz + d for ∀ z ∈ A is analytic. Whence, the mapping

f : A

→C determined by f ( z) =

az + b

cz + d for

∀ z

∈A is antianalytic by (P1).

Let f ( z) = u( x, y) + iv( x, y). Calculation shows that

det

∂u

∂ x

∂u

∂ y∂v

∂ x

∂v

∂ y

= ǫ

∂u

∂ x

2

+

∂v

∂ y

2 ,

where ǫ = 1 if f is analytic and −1 if f is antianalytic. This fact implies that an analytic

function preserves orientation but that an antianalytic one reverses the orientation.

4.4.2 Klein Surface. A Klein surface is a topological surface S together with a family

Σ = (U i, φi) | i ∈ Λ such that

(1) U i | i ∈ Λ is an open cover of S ;

(2) φi : U i → Ai is a homeomorphism onto an open subset Ai of C or C+;

(3) the transition functions of Σ defined in the following are dianalytic:

φi j = φiφ− j : φ j(U i

U j) −→ φi(U i

U j), i, j ∈ Λ.



Sec.4.4 NEC Groups 147

Usually, the family Σ is called to be an atlas and each (U i, φi) a chart on S , which is

positive if φi(U i) ⊂ C+. The boundary of S is determined by

∂S =

x

∈S

|there exists i

∈I , x

∈U i, φi( x)

∈R and φi(U i)

⊆C+

.

Particularly, if each transition function φi j is analytic, such a Klein surface is called a

Riemann surface in literature. Denote respectively by k (S ), g(S ) and χ(S ) the number

of connected components of ∂S , the genus and the Euler characteristic of S , where if

∂S ∅, we define its genus g(S ) to be the genus of the compact surface obtained by

attaching a 2-dimensional disc B2

to each boundary component of S . Then by applying

Theorem 4.2.6, we know the following result.

Theorem 4.4.1 Let S be a Klein surface. Then

χ(S ) =

2 − 2g(S ) − k (S ) if S is orientable,

2 − g(S ) − k (S ) if S is non − orientable.

Proof Let S be a surface without boundary, i.e., ∂S = ∅ with a definite triangulation.

We remove the interior of one triangle T to form a new surface S ′. Clearly, V (S ′) =

V (S ), E (S ′) = E (S ) and F (S ′) = F (S ) \ T . Whence, χ(S ′) = χ(S ) − 1. Continuous

this process, we finally get that χ(S ′) = χ(S ) − k if we remove k triangles on

S . Then we

know the result by Theorem 4.2.6.

Some important examples of Klein surfaces are shown in the following.

Example 4.4.2 Let H = z ∈ C | Im z > 0 and D = z ∈ C | | z| < 1 be respectively the

upper half plane and the unit disc in C shown in Fig.4.4.1 following.

¹

H

Im z

Re z

(a)

¹

| z| < 1

O

O

(b)

Fig.4.4.1

Choose atlas (U = H , φ = 1 H ) and (U = D, φ = 1 D) on H and D, respectively. Then




we know that both of them are Klein surfaces without boundary. Such Klein surfaces will

be always denoted by H and D in this book.

Example 4.4.3 The surface C+ with a structure induced by the analytic atlas

(C, 1C)

is

a Klein surface with boundary ∂C+ = R.

Example 4.4.4 LetC = C∪∞ and ∆ = C+ ∪∞. Then they are compact Klein surfaces

with atlas

Σ1 = (U 1 = C, φ1 = 1C), (U 2 = C 0, φ2 = z−1),

Σ2 = (U 1 = C+, φ1 = 1C+), (U 2 = ∆0, φ2 = z−1

),

respectively. Clearly, ∂C = ∅ and ∂∆ = R ∪∞.

4.4.3 Morphism of Klein Surface. Let A be a subset of C+, define A = z ∈ C | z ∈ A .

A folding mapping is the continuous mapping Φ : C → C+ determined by Φ( x + iy) =

x + i| y|. Clearly, Φ is an open mapping and Φ−1( A) = A ∪ A. Particularly, Φ−1(R) = R.

Let S and S ′ be Klein surfaces. A morphism f : S → S ′ from S to S ′ is a continuous

mapping such that

(1) f (∂S ) ⊆ ∂S ′;

(2) for ∀s ∈ S , there exist charts (U , φ) and (V , ψ) at points s and f (s), respectively

and an analytic function F : φ(U )

→C such that the following diagram

U V ¹

φ(U )

¹ C ¹ C+

f

F Φ

φ ψ (4 − 2)

commutes. It should be noted that in the case of Riemann surfaces, we only deal with

orientation-preserving morphisms, in which the diagram (4−2) is replaced by the diagram

(4 − 3) following.

U V ¹

φ(U )

¹ ψ(V )

(4 − 3)

F

f

φ ψ




Let S and S ′ be Klein surfaces and f : S → S ′ a morphism. If f is a homeomor-

phism, then S and S ′ are called to be isomorphic. Such a morphism f is isomorphism

between S and S ′. Particularly, if S = S ′, such a f is called automorphism of a Klein sur-

face S . Similarly, all automorphisms of S form a group with respect to the compositionof automorphisms, denoted by AutS . We present an example of automorphisms between

Klein surfaces following.

Example 4.4.5 Let H and D be Klein surfaces constructed in Example 4.4.2 and a map-

ping by ρ( z) = ( z + i)/(iz + 1). Then ρ : D → H is well-defined because if z = x + iy ∈ D,

so there must be x2 + y2 < 1 and consequently

ρ( z) =2 x + i(1 − x2 − y2)

x2 + (1 − y)2∈ H .

Furthermore, it is analytic, particularly continuous by definition. For s ∈ D, we choose

(U = D, 1 D) and (V = H , 1 H ) to be charts at s ∈ D and ρ(s) ∈ H , respectively. Then

Φ ρ = ρ for ρ( D) ⊂ H ⊂ C+ and the following diagram is commute.

U V ¹

φ(U )

¹ C ¹ C+

ρ

F = ρ Φ

1U 1V

Whence, ρ is a morphism between from Klein surfaces D to H . Now if g : H → C is

defined by g( z) =z − i

1 − iz, then g ρ = 1 H . Because ρ is onto, Img ⊂ D and ρg = 1 H , we

know that ρ is an isomorphism of Klein surfaces.

4.4.4 Planar Klein Surface. Let H = z ∈ C | Im z > 0 be a planar Klein surface

defined in Example 4.4.2 and let PGL(n,G) be the subgroup of GL(n,R) determined by

all A ∈ GL(n,R) with Det A 0. Now for A =

a b

c d

∈ PGL(2,R) with real entries,

we associate a mapping f A : H → H determined by

f A( z) =

az + b

cz + d if Det A > 0,

az + b

cz + d if Det A < 0.

Clearly, f A ∈ Aut H and f A = f cA for any non-zero c ∈ R. Hence, the mapping A → f A

embeds PGL(2,R) in Aut H . We prove this mapping is also surjective. In fact, let f ∈




Aut H and let ρ : D → H be the isomorphism determined in Example 4.4.5. Notice that

f is analytic, and so the same holds true for g = ρ−1 f ρ. Applying the maximum

principle of analytic function, g( z) =z − α

1

−α z

for some α ∈ D, µ ∈ C with | µ| = 1. Hence,

f ( z) =az + b

cz + d for some a, b, c, d ∈ C.

Because f ( H ) = H , we know that f (R \ −d /c) ⊂ R by continuity, and it is easy to

see that we can choose real numbers a, b, c, d . Notice that f (i) ∈ H implies that Det A =

ad − bc > 0.

If f reverses the orientation, let h : H → H be a mapping determined by h( z) =

− f ( z). Notice that h is an automorphism of H , i.e., h ∈ Aut H and it preserves the orienta-

tion. We know that

f ( z) =az + b

cz + d for some a, b, c, d ∈ R with Det A = ad − bc < 0.

Whence, we get the following result for the automorphism group of H .

Theorem 4.4.2 Let H = z ∈ C | Im z > 0 . Then

(1) Aut H = PGL(2,R);

(2) Aut H is a topological group, i.e., Aut H is both a topological space and a group

with a continuous mapping

∀ f

g−1 for f , g

∈Aut H.

4.4.5 NEC Group. A subgroup Γ of Aut H is said to be discrete if it is discrete as a

topological subspace of Aut H . Such a discrete group Γ is called to be a non-Euclidean

crystallographic group (shortly NEC group) if the quotient space H /Γ is compact.

Notice that there exist just two matrixes A, B ∈ GL(2,R) such that f A, f B for any

f ∈∈ Aut H with |Det A| = |Det B| = 1, i.e., B = − A, Det A = −Det A and Tr B = −Tr A.

Define Det f = Det A and Tr f = Tr A, respectively. Then we classify f ∈ Aut H into 3

classes with conditions following:

Hyperbolic. Det f = 1 and |Tr f | > 2.Elliptic. Det f = 1 and |Tr f | < 2.

Parabolic. Det f = 1 and |Tr f | = 2.

Furthermore, f is called a glide refection if Det f = −1, |Tr f | 0 or a refection if

Det f = −1, |Tr f | = 0. Denote by Aut+ H the subgroup of Aut H formed by all orien-

tation preserving elements in Aut H . Then it is clear that [Aut H : Aut+ H ] = 2. Call




an NEC group Γ to be Fuchsian if Γ ≤ Aut+ H . Otherwise, a proper NEC group. For

any NEC group Γ, the subgroup Γ+ = Γ ∩ Aut+ H is always a Fuchsian group, called the

canonical Fuchsian subgroup.

Calculation shows the following result is hold.

Theorem 4.4.3 Extend each f A ∈ Aut H to f onC ∪ ∞ in the natural way for A = a b

c d

∈ PGL(2,R) by

f A( z) =

−d /c i f z = ∞,

∞ i f z = −d /c,

az + b

cz + d i f Det f A = 1, z ∞, −d /c,

az + bcz + d

i f Det f A = −1, z ∞, −d /c.

Let f ∈ Aut H and Fix f = z ∈ C ∪∞| f ( z) = z. Then

Fix f =

two points on R ∪∞ i f f is hyperbolic or glide re f ection,

one point on R ∪∞ i f isparabolic,

two non − real con jugate points i f f is elliptic,

a circle or a line perpendicular to R i f f is a ref lection.

Let Γ be an NEC group. A fundamental region for Γ is a closed subset F of H

satisfying conditions following:

(1) If z ∈ H , then there exists g ∈ Γ such that g( z) ∈ F ;

(2) If z ∈ H and f , g ∈ Γ verify f ( z), g( z) ∈ IntF , then f = g;

(3) The non-Euclidean area of F \ IntF is zero, i.e.,

µ(F \ IntF ) =

F \IntF

dxdy

y2= 0.

The existence of fundamental region for an NEC group can be seen by the followingconstruction for the Dirichlet region with center p.

Construction 4.4.1 Let Γ be an NEC group. We construct its fundamental region in the

following. First, we show that there exists a point p ∈ H such that g( p) p for 1Γ g ∈ Γ.

In fact, we can assume the existence of an upper half Euclidean line l perpendicular to R

such that l Fix(γ ) for every γ ∈ Γ. Otherwise, we can get a sequence xn|n ∈ N




convergent to a point a ∈ H , lying on a Euclidean line parallel to R, and the upper half

Euclidean line ln perpendicular to R and passing through xn verifies ln = Fix(γ n) for some

γ n ∈ Γ. Consequently, γ n γ m if n m and limγ n(a) = limγ n( xn) = lim xn = a,

contradicts to the continuity of the mapping o : Aut H × H → H determined by o( f , x) =

f ( x) for f ∈ Aut H , x ∈ H .

Choose a sequence yn|n ∈ N of points H lying on l convergent to some point b ∈ H .

By assumption, there exists a sequence of pairwise distinct transformations gn|n ∈ N ⊂ Γ

such that gn( yn) = yn for every n ∈ N, which leads to a contradiction as before.

Now it is easy to check that

F = F p = z ∈ H |d ( z, p) ≤ d (g( z), p) for each g ∈ Γ

is a fundamental region of Γ, where d (u, v) is the non-Euclidean distance between uand v, i.e.,

d (u, v) =

C u,v

(dx2 + dy2)1/2

y,

C u,v being the geodesic joining u and v, i.e., a circle or a line orthogonal to R. Then F p

verifies conditions (1)-(3):

(1) Let z be a point in H . Since Γ is discrete, the orbit O z of z under Γ is closed. Thus

there exists w ∈ O z such that d (w, p) ≤ d (w′, p) for each w′ ∈ O z. If w = g( z), g ∈ Γ, then

it is clear that g( z) = w

∈F p.

(2) Obviously that

IntF p = z ∈ H |d ( z, p) < d (g( z), p), for each g ∈ Γ \ 1 H .

Then z ∈ H , f , g ∈ Γ and f ( z), g( z) ∈ IntF p imply that for f g,

d ( f ( z), p) < d (g f −1( f ( z), p)) = d (g( z), p), d (g( z), p) < d ( f g−1(g( z), p)) = d ( f ( z), p),

a contradiction. Thus, f = g.

(3) This is follows easily from the fact that the boundary of F p

is a convex polygon

with a finite number of sides in the non-Euclidean metric.

Usually, a fundamental region F of an NEC group verifying conditions following is

called regular :

(1) F is a bounded convex polygon with a finite number of sides in the non-Euclidean

metric;




(2) F is homeomorphic to a closed disc;

(3) F \ IntF is a closed Jordan curve and there are finite vertices on F \ IntF which

divide it into the following classes e of Jordan arcs:

(3.1) e = F ∩ gF , where g ∈ Γ is a reflection;

(3.2) e = F ∩ gF , where g ∈ Γ, g2 1 H ;

(3.3) e for which there exists an elliptic transformation g ∈ Γ, g2 = 1Γ such that

e ∪ ge = F ∩ gF ;

(4) If F , gF do not have an edge in common for a g ∈ Γ, then F ∩ gF has just one

point.

Then we know the following conclusion.

Theorem 4.4.4 For any NEC group Γ , there exist regular fundamental regions, such as

F p for example.

Construction 4.4.2 Let F be a regular fundamental region of an NEC group Γ. For a

given g ∈ Γ, gF is said to be a face. Clearly, the mapping Γ → faces determined by

g → gF is a bijection and H =g∈Γ

gF . In fact, gF |g ∈ Γ is a tessellation of H .

(1) Given a side e of F , let ge be the unique transformation for which geF meets F

in the edge e, i.e., e = F ∩ geF . then ge|e ∈ sides of Γ is a set of generators of γ . In

fact, for ∀g ∈ Γ there exists a sequence of elements g1 = 1 H , g − 2, · · · , gn+1 in Γ such

that giF meet gi+1F one to another in a side, say gi(ei), where ei is a side of F . Clearly,

gi(geif ) = gi+1F and so gi+1 = gigei

for 1 ≤ i ≤ n. Consequently, g = ge1ge2

· · · genfor

some sides e1, e2, · · · , en of F .

(2) First, we label sides of type (3.1). Afterward, if we label e a side of type (3.2)

or (3.3), the side ge is labeled e′ if g ∈ Γ+, and e∗ if g ∈ Γ \ Γ+. We write down the labels

of the sides in counter-clockwise order and say (e, e′), (e, e∗) pair sides. In this way, we

obtain the surface symbols, which enables one to determine the presentation of Γ and the

topological structure H /Γ, such as those claimed in Theorem 4.2.2.

(3) Let a and a be pair sides and let g ∈ Γ be an element such that g−1(a) = a.

For a hyperbolic arc f joining two vertices of F and splitting F into two regions A and B

containing a anda, respectively, A∪gB is a new fundamental region of Γ which has a new

pair sides b andb withb = g−1(b) instead of a anda and suitably relabeled other sides.

Repeating this procedure in suitable way one can arrive to a fundamental region with the




following side labelings

ξ 1 ξ ′1 · · · ξ r ξ ′r ǫε1γ 10 · · · γ 1s1ε′

1 · · · εk γ k 0 · · · γ ksk ε′

k α1 β1α′1 β

′1 · · · α p β pα′

p β′ p (4 − 4)

ξ 1 ξ ′1 · · · ξ r ξ ′r ǫε1γ 10 · · · γ 1s1ε′

1 · · · εk γ k 0 · · · γ ksk ε′

k δ1δ′1 · · · δqδ′

q (4 − 5)

according to H /Γ orientable or not.

(4) Identify points on pair side, we get that H /Γ is a sphere with k disc removed and

p handles or q crosscups added if (4 − 3) or (4 − 4) holds.

(5) For getting the defining relations for Γ, consider the faces meeting at each vertex

of F . Notice that Γ is discrete. The number of these faces is finite. Choose one of vertices

of Γ and let l = L0, L1, · · · , Ln, Ln+1 = L be the corresponding chain faces. Obviously,

there exist g1, · · · , gn of elements of Γ such that

L1 = g1 L, L2 = g2g1 L, · · · , L = Ln+1 = gn · · · g1 L.

Whence, every vertex induces a relation

gngn−1 · · · g2g1 = 1 H .

It turns out that these relations of this type and g2e = 1 H coming from such sides of F fixed

by a unique nontrivial element ge

∈Γ form all defining relations of Γ.

(6) As we get a surface symbol (4 − 4) or (4 − 5) and using procedures described in

(1) and (5), we find the presentation of Γ following:

Generators: xi, 1 ≤ i ≤ r ;

ei, 1 ≤ i ≤ k ;

ci j, 1 ≤ i ≤ k , 1 ≤ j ≤ si;

ai, bi, 1 ≤ i ≤ p in the case (4 − 4);

d i, 1 ≤ i ≤ q in the case (4 − 5).

Relations: xm−i

i = 1Γ, 1 ≤ i ≤ r ;

e−1i

ci0eicisi= 1Γ, 1 ≤ i ≤ k ;

c2i, j−1 = c2

i j = (ci, j−1ci j)ni j = 1;

x1 · · · xr e1 · · · ek [a1, b1] · · · [a p, b p] = 1 in case (4 − 4);

x1 · · · xr e1 · · · ek d 21· · · d 2q = 1 in case (4 − 5),




where a, b, c, d , e, x correspond to these transformations induced by edges α, β, γ, δ, ε, ξ ,

[ai, bi] = aibia−1i

b−1i and mi, n j are numbers of faces meeting F at common vertices for

sides ( ξ i, ξ ′i ) and (γ i, j−1, γ i j), respectively.

For an NEC group Γ with the previous presentation, we define the signature σ(Γ) of Γ by

σ(Γ) = (g; ±; [m1, · · · , mr ]; (n11, · · · , n1s1), · · · , (nk 1, · · · , nksk

)),

and its hyperbolic area µ(Γ) by

µ(Γ) =

αg + k − 2 +

r i=1

(1 − 1

mi

) +1

2

k i=1

si j=1

(1 − 1

ni j

)

,

where g = p, the sign + and α = 2 in (4 − 4) or g = q, the sign − and α = 1 in (4-5), i.e.,

orientable in the first and non-orientable otherwise. It has been shown that µ(Γ) is just thehyperbolic area of the fundamental of Γ and independent on its choice.

Usually, if r = 0, si = 0 or k = 0, we denote these [m1, · · · , mr ], (ni1, · · · , nisi) by [−],

(−) or −, respectively. For example,

σ(Γ) = (g; ±; [−]; (−), · · · , (−) k

)

if r = 0 and si = 0. Such an NEC group is called to be a surface group. Partic-

ularly, if k = 0, i.e., these fundamental groups in Theorem 4.3.10, the signature is

σ(Γ) = (g; ±; [−]; (−)). Clearly, the area of a surface group Γ is µ(Γ) = 2π(αg + k − 2).

Theorem 4.4.5(Hurwitz-Riemann formula) Let Γ be a NEC subgroup of a NEC group

Γ′. Then µ(Γ)

µ(Γ′)= [Γ′ : Γ].

Proof Notice that Γ is a discrete as a subgroup of Γ′. By definition, H /Γ′ and H /Γ

are compact, so Γ′ and Γ have compact fundamental regions F ′ and F . Let h1, · · · , hk ∈ Γ′

be the coset representatives of Γ, where k = [Γ′ : Γ]. Then It is easily to know that

F = h1(F ′) ∪ · · · ∪ hk (F ′). Consequently,

µ(Γ) = area(F ) =

k i=1

area(hi(F ′)) = k × area(F′) = k × µ(Γ′).

Thus, µ(Γ)

µ(Γ′)= [Γ

′ : Γ].




$4.5 AUTOMORPHISMS OF KLEIN SURFACES

4.5.1 Morphism Property. We prove the automorphism group of a Klein surface is finite

in this section. For this objective, we need to characterize morphisms of Klein surfaces inthe first.

Theorem 4.5.1 Let f : S → S ′ be a non-constant morphism and (U , φ), (V , ψ) two

charts in S and S ′ with f (U ) ⊂ V, ψ(V ) ⊂ C+. Then there exists a unique analytic

mapping F : φ(U ) → C such that the following diagram

U V ¹

φ(U )

¹ C ¹ ψ(V )

f

F Φ

φ ψ

commutes.

Proof First, if there are two non-constant analytic mappings F , F ′ : φ(U ) → C such

that ΦF = ΦF ′, then F = F ′ or F = F ′. Let Y ⊂ F −1(C \ R) be a nonempty connected

set. Choose M 1 = x ∈ Y |F ( x) = F ′( x) and M 2 = x ∈ Y |F ( x) = F ′( x). Then M 1 and M 2

are closed and disjoint with Y = M 1 ∪ M 2, which enables one to get M 1 = Y or M 2 = Y .

If M 2 = Y , F must be both analytic and antianalytic on Y . Thus F |Y is constant, and so F

is constant by the properties of analytic functions, a contradiction. Whence, F = F ′.

Now suppose that we can cover U by U j| j ∈ J such that there are analytic mappings

F j : φ(U j) → C with the following diagram

U V ¹

φ(U )

¹

C¹

ψ(V )

f

F j Φ

φ ψ

commutes. Then these mappings F j glue together will produce a function F that we are

looking for. So we only need to find such mappings F j.

By definition, for x ∈ U and y = f ( x) ∈ V , there exist charts (U x, φ x and (V y, ψ y) and

an analytic mapping F x with U x ⊂ U , V y ⊂ V such that the following diagram commutes:



Sec.4.5 Automorphisms of Klein Surfaces 157

U x V y¹

φ x(U x)

¹ C ¹ ψ y(V y)

f

F x Φ

φ x ψ y

We construct a mapping F ∗ x such that the following diagram also commutes:

U x V y¹

φ x(U x

)

¹

C¹

ψ y(V y

)

f

F ∗ x Φ

φ x ψ y

In fact, for any given u ∈ φ(U x), we know that F xφ xφ−1(u) ∈ Φ−1(Imψ y) = ψ y(V y)∪ψ y(V y).

Consider (ψψ−1 y )∧ : ψ y(V y) ∪ ψ y(V y) → C. Then according with φ xφ−1 and ψψ−1

y were

analytic or antianalytic, we take F ∗ x or F ∗ x to be (ψψ−1 y )∧F xφ xφ−1. Then we get such F j as

one wish.

A fundamental result concerning the behavior of morphisms under composition is

shown in the following.

Theorem 4.5.2 Let S , S ′ and S ′′ be Klein surfaces and f : S → S ′ , g : S ′ → S ′′

continuous mappings such that f (∂S ) ⊂ ∂S ′ , g(∂S ′) ⊂ ∂S ′′. Consider the following

assertions:

(1) f is a morphism;

(2) g is a morphism;

(3) g f is a morphism.

Then (1) and (2) imply (3). Furthermore, if f is surjective, (1) and (3) imply (2), and if f

is open, (2) and (3) imply (1).

The proof of Theorem 4.5.2 is not difficult. Consequently, we lay it to the reader as

an exercise.

Corollary 4.5.1 Let S and S ′ be topological surfaces and f : S → S ′ a continuous

mapping. Then




(1) If S ′ is a Klein surface, then there is at most one structure of Klein surface on S

such that f is a morphism.

(2) If f is surjective and S is a Klein surface, then there exists at most one structure

of Klein surface on S ′ such that f is a morphism.

4.5.2 Double Covering of Klein Surface. Let S be a Klein surface with atlas

=

(ui, φi)|i ∈ I . Suppose S is not a Riemann surface and define

U ′i = U i × i × 1 and U ′′i = U i × i × −1,

where i runs over I . We identify some points in

X = i∈ I

U ′i i∈ I

U ′′i .

(1) For i ∈ I and Di = ∂S ∩ U i, identify Di × i × 1 with Di × i × −1.

(2) For ( j, k ) ∈ I × I such that U j meets U k , let W be a connected component in

U j ∩ U k . Identify W × j × δ with W × k × δ for δ = ±1 if φ jφ−1k

: φk (W ) → C

is analytic, and W × j × δ with W × k × −δ for δ = ±1 if φ jφ−1k

: φk (W ) → C is

antianalytic.

Put S C = X /identificationsabove. For each i ∈ I , let φ′i : U ′

i→ C determined by

φ′i ( x, i, 1) = φi( x) and φ′′

i : U ′′i

→C determined by φ′

i ( x, i,

−1) = φi( x). Obviously, if

p : X → S C denotes the canonical projection and U i = p(U ′i ∪ U ′′

i ), the family U i|i ∈ I is an open cover of S C . Furthermore, each mapping φi : U i → C defined by φi(u) = φ′(u)

if u ∈ U ′i or φi(u) = φ′′(u) if u ∈ U ′′

i is a homeomorphism onto its image. Thus

C =

(U i,φi|i ∈ I ) is an analytic atlas on S C . Clearly, ∂S C = ∅. Whence, S C is a Riemann

surface by construction.

We claim that there exists a morphism f : S C → S and an antianalytic mapping

σ : S C → S C such that f σ = f and σ2 = 1S . In fact, it is suffices to determine

f : S C → S by f : u = p(v, i, δ) → v for v ∈ U i and δ = ±1. It should be noted that each

fibers of f has one or two points and we define

σ : S C → S C : u → u if | f −1( f (u))| = 1,

f −1( f (u)) if | f −1( f (u))| = 2.

Such a triple (S C , f , σ) is called the double cover of S .

We know the following result due to Alling-Greenleaf ([BEGG]):




Theorem 4.5.3 Let g be a morphism from a Riemann surface S onto a Klein surface S ′

with the double cover (S ′C

, f ′, σ). Then there exists a unique morphism g′ : S → S ′C

such

that f ′g′ = g.

4.5.3 Discontinuous Action. Let S be a Klein surface and G ≤ AutS . We say G

acts discontinuously on S if each point x ∈ S possesses a neighborhood U such that

GU is finite. Furthermore, G is said to be acts properly discontinuously on S if it acts

discontinuously on S satisfying conditions following:

(1) For ∀ x, y ∈ S with x yG, there are open neighborhoods U and V at points x

and y such that there are no f ∈ G with U ∩ f (V ) ∅;

(2) For x ∈ S , 1S f ∈ G x and the mapping φ x f φ−1 x is analytic restricted suitably, x

is isolated in Fix( f ).

For the existence of properly discontinuously groups, we know the following result

as an example.

Theorem 4.5.4 Every discrete subgroup Γ of Aut H acts properly discontinuously on H.

Proof First, the stabilizer Γ of each x ∈ H is finite. Otherwise, let f n|n ∈ Z + ⊂ Γ x

such that f n f m if n m and so limn→∞

f n( x)|n ∈ Z + = x. But then Γ must be not discrete.

Now let N be the set of natural numbers m such that H contains the Euclidean ball

Bm with center x and radius 1/m. Let Γm = Γ Bm. Then there must be

Γ x =n∈ Z +

Γm.

In fact, if f Γ x, take open disjoint neighborhoods U and V of x and f ( x). If m is bigger

enough, Bm ⊂ U , f ( Bm) ⊂ V . Thus there must be f Γm. On the other hand, if f ∈ Γ x,

then there is an integer m0 such that for any integer m ≥ n0, Bm = f ( Bm). This establishes

the previous equality.

(1) Γ acts discontinuously on H . Assume that each Γm is infinite. Then the finiteness

of Γ x and the above equality imply that

Γm1 Γm2

· · ·

for some sequence mk |k ∈ Z + ⊂ Z +. Choose f k ∈ Γmk \ Γmk +1

. Clearly, f k f l if k l.

However, if we take x ∈ Bmk ∩ f k ( Bmk

) and y ∈ Bmk with xk = f ( yk ), then

limk →∞

xk |k ∈ Z + = x = limk →∞

yk |k ∈ Z +.




So limk →∞

f ( xk )|k ∈ Z + = x, which contradicts the discreteness of Γ.

(2) For x, y ∈ H , x yAut H , there are neighborhoods U of x and V of y such that

there are no f ∈ G with U ∩ f (V ) ∅. In fact, let P be the set of numbers m ∈ Z + such

that the balls Bm and B′m of radius 1/m with centers x and y, respectively, are containedin H . We prove that there are no f ∈ Γ with Bm ∩ f ( B′

m) ∅ for all m ∈ P. Denoted by

Dm = f ∈ Γ| Bm ∩ f ( B′m) ∅. Clearly,

m∈P

Dm = ∅. Otherwise, for some f ∈ Γ there

are points xm ∈ Bm and ym ∈ B′m with f ( ym) = xm, m ∈ P, which implies f ( y) = x, i.e.,

x ∈ yAut H , a contradiction. So we have

Dm1 Dm2

· · ·

for some sequence mk |k ∈ Z + ⊂ P. Choose f k ∈ Dmk \ Dmk +1

. then we know that

limk →∞ f k ( y)|k ∈ Z + = x, f k f l if k l, contradicts the discontinuousness of Γ.

(3) Given 1 H f ∈ Γ, f has the form

f ( z) =az + b

cz + d , (b, c, d − a) (0, 0, 0).

Thus Fix( f ) \ x is finite, i.e., x is isolated in Fix( f ).

The importance of these properly discontinuously groups on Klein surfaces is im-

plied in the next result.

Theorem 4.5.5 Let G be a subgroup of AutS which acts properly discontinuously on theKlein surface S . Then S ′ = S /G admits a unique structure of Klein surface such that

π : S → S ′ is a morphism.

A complete prof of Theorem 4.5.5 can be found in [BEGG1]. Applying Theorems

4.5.4 and 4.5.5 to the planar Klein surface H , we know the following conclusion.

Theorem 4.5.6 For a discrete subgroup Γ of Aut H, the quotient H /Γ admits a unique

structure of Klein surface such that the canonical projection H → H /Γ is a morphism of

Klein surfaces. Particularly, this holds true if Γ is an NEC group.

Generally, we also know the following result with proof in [BEGG1], which enables

one to find Klein surfaces on topological surfaces with genus≥ 3.

Theorem 4.5.7 If S is a Klein surface and 2g(S ) + k (S ) ≥ 3 if S is orientable, or

g(S ) + k (S ) ≥ 3 otherwise. Then there exists a surface NEC group Γ such that S and

H /Γ are isomorphic Klein surfaces and S C = H /Γ+ , where Γ+ is a subgroup formed by




orientation preserving elements in Γ. In fact, |Γ : Γ+| = 2. Furthermore, if π′ : H → H /Γ

be the canonical projection, i.e, Γ = f ∈ Aut H |π′ f = π′.

According to this theorem, we can construct Klein surfaces on compact surfaces S

unless S is the sphere, torus, projective plane or Klein bottle.

4.5.4 Automorphism of Klein Surface. Let S and S ′ be compact Klein surfaces. Denote

by Isom(S ′, S ) all isomorphisms from S ′ to S . If they satisfy these conditions in Theorem

4.5.6, then they can be represented by H /Γ′, H /Γ for some NEC group Γ′ and Γ. Let

π : H → H /Γ and π′ : H → H /Γ′ be the canonical projections and

A(Γ′, Γ) = g ∈ Aut H |π′( x) = π′( y) if and only if πg( x) = πg( y).


Theorem 4.5.8 Let g ∈ Aut H. The following statements are equivalent:

(1) g ∈ A(Γ′, Γ);

(2) there is a uniqueg ∈ Isom( H /Γ′, H /Γ) with the following commutative diagram:

H H

S ′ S

¹

¹

g

gπ′ π

(3) Γ′ = g−1Γg.

Proof (1) ⇒ (2). For x′ = π′( x) ∈ S ′, define g( x′) = gπ′( x) = πg( x). Applying

Theorem 4.5.2, we know thatg is a homeomorphism on H by the definition of A(Γ, Γ′).

(2) ⇒ (3). Applying Theorem 4.5.7, if f ∈ Γ′ and h = g f g−1, then

πh = πg f g−1 =gπ′ f g−1 =gπ′g−1 = πgg−1 = π,

i.e., h ∈ Γ and so Γ′ ⊂ g−1Γg. Conversely, if h ∈ g−1Γg, then ghg−1 ∈ Γ, i.e., πghg−1 = π.

Sogπ′h =gπ′. Notice thatg is bijective. We know π′h = π′, i.e., h ∈ Γ.

(3) ⇒ (1). Let x, y ∈ H with π′( x) = π′( y) and y = f ( x) for some f ∈ Γ′ = g−1Γg.

Now h = g f g−1 ∈ Γ. Notice that hg = g f and πh = π. We find that

π(g( y)) = π(g( f ( x))) = π(h(g( x))) = π(g( x)).

The converse is similarly proved.




Theorem 4.5.9 Let S = H /Γ and S ′ = H /Γ′. Then

(1) S and S ′ are isomorphic if and only if Γ and Γ′ are conjugate in Aut H.

(2) AutS

≃N Aut H (Γ)/Γ , where N Aut H (Γ) is the normalizer of Γ in Aut H.

Proof Obviously, S and S ′ are isomorphic if and only if A(Γ, Γ′) ∅. By Theorem

4.5.8, we get the assertion (1).

For (2), we prove first that the mapping A(Γ, Γ′) → Isom(S ′, S ) is surjective. In

fact, if S and S ′ are Riemann surfaces, let φ ∈ Isom(S ′, S ) and ( H , π) and ( H ′, pi′) be

the universal coverings of S and S ′, respectively. Then by the Monodromy theorem and

Theorem 4.5.2, there exists g ∈ Aut H such that the following diagram is commutative.

H H

S ′ S

¹

¹

g

φ

π′ π

It is clear that g ∈ A(Γ, Γ′). So φ =g by Theorem 4.5.8.

Generally, let f : S C → S and f ′ : S ′C

→ S ′ be the double coverings with the

corresponding antianalytic involutions σ : S C → S C and σ′ : S ′c → S ′

C . By Theorem

4.5.3, there exists ψ ∈ Isom(S ′C , S C ) such that the following diagram

S ′C S C

S ′ S

¹

¹

φ

φ

f ′ f

is commutative. Let p : H → S C and p′ : H → S ′C be the canonical projections. As we

shown for Riemann surfaces, there exists g ∈ Aut H such that the following diagram

H H

S ′C S C

¹

¹

g

φ

p′ p

is commutative. Now up to the identifications of S with H /Γ and S ′ with H /Γ′, the

mappings π′ = f ′ p′ : H → S ′ and π = f p : H → S are the canonical projections, which

enables us to obtain a commutative diagram following.






Theorem 4.5.11 Let Γ be an NEC group. Then N Aut H (Γ) in Aut H is also an NEC group.

Proof Notice π : H → H /Γ. We immediately find the compactness of H / N Aut H (Γ)

from H under π. Because Aut H is a topological group, we only need to check that the

identity 1 H is an open subset in N Aut H (Γ).

We claim that there exist 1 H h1, h2 ∈ Γ+ such that Fix(h1) Fix(h2). In fact, let

h1 ∈ Γ+ defined by h1( z) = r 0 z for some r 0 ∈ R. Then Fix(h1) = 0, ∞. If there are

another h ∈ Γ+, h h1 such that Fix(h) = 0, ∞, then

Γ+ ⊂ A = f : H → H | f ( z) = rz, r ∈ R+, z ∈ C.

Since H /Γ+ is compact, the same holds for H / A ≈ (0, 1), a contradiction.

Now let C Aut H (h1, h2) =

h

∈Aut H

|hhi = hih, i = 1, 2

. We prove that C Aut H (h1, h2)

is trivial. Applying Theorem 4.5.10, if there are 1 H h ∈ C Aut H (h1, h2) ∩ Aut+ H ,

then Fix(h1) = Fix(h) = Fix(h2), a contradiction. On the other hand, if there are h ∈C Aut H (h1, h2) \ Aut+ H , then h2 = 1 H , and so h( z) = − z. Now hhi = hih implies that

hi( z) = −1/ z for i = 1, 2, also a contradiction. Thus the mapping ζ i : N Aut H (Γ) → Γ by

g → ghig−1 are well-defined and continuous with ζ i(1 H ) = hi.

Since Γ is discrete, we can find open neighborhoods V 1, V 2 of 1 H in N Aut H (Γ) such

that ζ i(V i) ⊂ hi, i.e., ghig−1 = hi, i = 1, 2 for each g ∈ V = V 1 ∩ V 2. In other words,

V

⊂C Aut H (h1, h2) =

1 H

. Thus

1 H

= V is open in N Aut H (Γ).

A group of automorphism of a Klein surface S is a subgroup of AutS . We get the

following consequence by Theorem 4.5.11.

Corollary 4.5.2 A group G ≤ AutS with S = H /Γ if and only if G ≃ Γ′/Γ for some NEC

group Γ′ with Γ⊳ Γ′.

Proof Applying Theorem 4.5.11, G is a subgroup of N Aut H (Γ)/Γ. So there is a

subgroup Γ′ of N Aut H (Γ) containing Γ such that H /Γ′ is compact. Notice Γ′ is also discrete.

Whence, Γ′ is a NEC group.

Now we prove the main result of this section.

Theorem 4.5.12 Let S be a compact Klein surface with conditions in Theorem 4.5.7 hold.

Then AutS is finite.

Proof Let S = H /Γ. By Theorem 4.5.10, N Aut H (Γ) is an NEC group. Applying

Theorem 4.4.5, we know AutS is finite by that of the group index [ N Aut H (Γ) : Γ].



Sec.4.6 Remarks 165

$4.6 REMARKS

4.6.1 Topology, including both the point topology and the algebraic topology has become

one of the fundamentals of modern mathematics, particularly for geometrical spaces.

Among them, the simplest is the surfaces fascinating mathematicians in algebra, geome-

try, mathematical analysis, combinatorics, · · ·, and mechanics. There are many excellent

graduated textbooks on topology, in which the reader can find more interested materials,

for examples, [Mas1]-[Mas2] and [Mun1].

4.6.2 Similar to Theorem 4.2.4 on compact surface without boundary, we can classify

compact surface with boundary and prove the following result.

Theorem 4.6.1 Let S be a connected compact surface with k

≥1 boundaries. Then its

surface presentation is elementary equivalent to one of the following:

(1) Sphere with k ≥ 1 holes

aa−1c1 B1c−11 c2 B2c−1

2 · · · ck Bk c−1k ;

(2) Connected sum of p tori with k ≥ 1 holes

a1b1a−11 b−1

1 a2b2a−12 b−1

2 · · · a pb pa−1 p b−1

p c1 B1c−11 c2 B2c−1

2 · · · ck Bk c−1k ;

(3) Connected sum of q projection planes with k ≥ 1 holes

a1a2 · · · aqc1 B1c−11 c2 B2c−1

2 · · · ck Bk c−1k .

4.6.3 The conception of fundamental group was introduced by H.Poincare in 1895. Sim-

ilarly, replacing equivalent loops of dimensional 1 based at x0 by equivalent loops of

dimensional d , we can extend this conception for characterize those higher dimensional

topological spaces with resemble structure of surface.

4.6.4 The conception of Klein surface was introduced by Alling and Greenleaf in 1971

concerned with real algebraic curves, correspondence with that of Riemann surface con-

cerned with complex algebraic curves (See [All1] for details). The materials in Sections

4.5.4 and 4.5.5 are mainly extracted from the reference [BEGG1]. Certainly, all Rie-

mann surfaces are orientable. Their surface group is usually called the Fuchsian group

constructed similarly to that of Construction 4.4.2. It should be noted that each surface



166 Chap.4 Surfaces Groups

in Construction 4.4.2 for an NEC group maybe with boundary. This construction also

establishes the relation of surfaces with that of NEC groups, enables one to research au-

tomorphisms of Kleins surface by that of combinatorial maps.



CHAPTER 5.

Map Groups

A map group is a subgroup of an automorphism group of map, which is also akind of geometrical group, i.e., a subgroup of triangle groups. There are two

ways for such groups in literature. One is by combinatorial techniques. An-

other is the classical by that of algebraic techniques. Both of them have their

self-advantages and covered in this chapter. The materials in Sections 5.1–

5.2 are an elementary introduction to combinatorial maps. By the discussion

of Chapter 4, we explain how to embed a graph and how to characterize an

embedding of graph on surface in Section 5.1, particularly these techniques

related to algebraic maps, such as those of rotation system, band decompo-

sition of surface, traveling ruler and orientability algorithm in Section 5.1.

This way naturally introduce the reader to understand the correspondence be-

tween embeddings and maps, and the essence of notations α, β and P, or

flags in an algebraic map (X α, P). The automorphisms of map with prop-

erties are discussed in Section 5.3, characterized by behavior of maps or the

semi-arc automorphism of its underlying graph. The materials in Sections

5.4–5.5 concentre on regular maps, both by combinatorial and algebraic tech-

niques, which are closely related combinatorics with geometry and algebra.

By explaining how to get a regular tessellation of a plane, a geometrical way

for constructing regular maps by triangle group is introduced in Section 5.5.

After generalizing the conception of surface to multisurface S in section 5.5,

we also show how to construct maps M on multisurfaces S such that the pro-

jection of M on each surface of S is a regular map.



168 Chap.5 Map Groups

§5.1 GRAPHS ON SURFACES

5.1.1 Cell Embedding. Let G be a connected graph with vertex set V (G) and edge set

E (G) and S a surface. An 2-cell embedding of G on S is geometrical defined to be a con-

tinuous 1−1 mapping τ : G → S such that each component in S −τ(G) homeomorphic to

an open 2-disk. Certainly, the image τ(G) is contained in the 1-skeleton of a triangulation

of the surface S . Usually, components in S − τ(G) are called faces. For example, we have

shown an embedding of K 4 on the sphere and Klein bottle in Fig.5.1.1(a) and Fig.5.1.1(b)

respectively.

¹

u3

2

2

(b)(a)

u1

u2

u4u3

u1 u2

1u4

1

Fig.5.1.1

For v ∈ V (G), denote by N eG(v) = e1, e2, · · · , e ρ(v) all the edges incident with the

vertexv

. A permutation one

1,e

2, · · · ,e

ρ(v) is said apure rotation

. All pure rotationsincident with v is denoted by (v). A pure rotation system of the graph G is defined to be

ρ(G) =

(v)|v ∈ V (G).

For example, the pure rotation systems for embeddings of K 4 on the sphere and Klein

bottle are respective

ρ(K 4) = (u1u4, u1u3, u1u2), (u2u1, u2u3, u2u4), (u3u1, u3u4, u3u2), (u4u1, u4u2, u4u3),

ρ(K 4) = (u1u2, u1u3, u1u4), (u2u1, u2u3, u2u4), (u3u2, u3u4, u3u1), (u4u1, u4u2, u4u3)

and intuitively, we can get a pure rotation system for each embedding of K 4 on a locally

orientable surface S .

In fact, there is a relation between these pure rotation systems of a graph G and its

embeddings on orientable surfaces S , called the rotation embedding scheme, observed

and used by Dyck in 1888, Heff ter in 1891 and then formalized by Edmonds in 1960

following.



Sec.5.1 Graphs on Surfaces 169

Theorem 5.1.1 Every embedding of a graph G on an orientable surface S induces a

unique pure rotation system ρ(G). Conversely, Every pure rotation system ρ(G) of a graph

G induces a unique embedding of G on an orientable surface S .

Proof If there is a 2-cell embedding of G on an orientable surface S , by the definition

of surface, there is a neighborhood Du on S for u ∈ V (G) which homeomorphic to a

dimensional 2 disc ϕ : Du → ( x1, x2) ∈ R 2| x21 + x2

2 < 1 such that each edge incident with

u possesses segment not in Du. Denoted by ∂ Du = ( x1, x2) ∈ R 2| x21 + x2

2 = 1 and let the

counterclockwise order of intersection points of edges uv, v ∈ N G(u) with that of ∂ Du be

pv1, pv2

, · · · , pv ρ(u). Define a pure rotation of u by (u) = (uv1, uv2, · · · , uv ρ(u)). Then we get

a pure rotation system ρ(G) =

(u), u ∈ V (G).

Conversely, assume that we are given a pure rotation system ρ(G). We show that this

determines a 2-cell embedding of G on a surface. Let D denote the digraph obtained by

replacing each edge uv ∈ G with (u, v) and (v, u). Define a mapping π : E ( D) → E ( D)

by π(u, v) = (v)(v, u), which is 1 − 1, i.e., a permutation on E ( D). Whence π can be

expressed as a product of disjoint cycles. Each cycle is an orbit of π action on D( E 0.

Thus the orbits partition the set E ( D). Assume

F : (u, v)(v, w) · · · ( z, u)

is such a orbit under the action of π, simply written as

F : (u, v, w, · · · , z, u).

Notice this implies a traveling ruler , i.e., beginning at u and proceed along (u, v) to v,

the next arc we encounter after (u, v) in a counterclockwise direction about v is ρ(v)(v, u).

Continuing this process we finally arrive at the arc ( z, u), return to u and get the boundary

of a 2-cell.

Let F 1, F 2, · · · , F l be all 2-cells obtained by the traveling ruler on E ( D). Applying

Theorem 4.2.2, we know it is a polygonal representation of an orientable surface S by

identifying arc pairs (u, v) with (v, u) in E ( D).

According to this theorem, we get the number of embeddings of a graph on orientable

surfaces following.

Corollary 5.1.1 The number of embeddings of a connected graph G on orientable sur-

faces is

v∈V (G)

( ρ(v) − 1)!.




5.1.2 Rotation System. For a 2-cell embedding of a graph G on a surface S , its embed-

ded vertex and face can be viewed as 0 and 2-disks, and its embedded edge can be viewed

as a 1-band defined as a topological space B with a homeomorphism h : I × I → B, where

I = [0, 1], the unit interval. The arcs h( I × i) for i = 0, 1 are called the ends of B, andthe arcs h(i × I ) for i = 0, 1 are called the sides of B. A 0-band or 2-band is just a

homeomorphism of the unit disk. A band decomposition of the surface S is defined to be

a collection B of 0-bands, 1-bands and 2-bands with conditions following hold:

(1) The diff erent bands intersect only along arcs in their boundary;

(2) The union of all the bands is S , i.e.,

B∈B B = S ;

(3) The ends of each 1-band are contained in a 0-band;

(4) The sides of each 1-band are contained in a 2-band;

(5) The 0-bands are pairwise disjoint, and the 2-bands are pairwise disjoint.

For example, a band decomposition of the torus is shown in Fig.5.1.2, which is an

embedding of the bouquet B2 on T 2.

O

¹

¹

e1

e2

Fig.5.1.2

A band decomposition is called locally orientable if each 0-band is assigned an ori-

entation. Then a 1-band is called orientation-preserving if the direction induced on its

ends by adjoining 0-bands are the same as those induced by one of the two possible orien-

tations of the 1-band. Otherwise, the 1-band is called orientation-reversing, such as thoseshown in Fig.5.1.3 following.

Orientation-preserving band Orientation-reversing band

Fig.5.1.3




An edge e in a graph G embedded on a surface S associated with a locally ori-

entable band decomposition is said to be type 0 if its corresponding 1-band is orientation-

preserving, and type 2, otherwise. A walk in this associated graph is type 1 if it has an

odd number of type 1 edges and type 0, otherwise.For such a graph G associated with a locally orientable band decomposition, we

define a rotation system ρ L(v) of v ∈ V (G) to be a pair ( J (v), λ), where J (v) is a pure

rotation system and λ : E (G) → Z2 is determined by λ(e) = 0 or λ(e) = 1 if e is type 0 or

type 1 edge, respectively. For simplicity, we denote the pairs (e, 0) and (e, 1) by e and e1,

respectively. The rotation system ρ L(G) of G is defined by

ρ L(G) = ( J (v), λ)|J (v) ∈ ρ(G), λ : E (G) → Z2.

For example, the rotation system of the complete graph K 4 on the Klein bottle shown in

Fig.5.1.1(b) is

ρ L(K 4) = (u1u2, u1u13, u1u4), (u2u1, u2u3, u2u4), (u3u2, u3u4, u3u1

1), (u4u1, u4u2, u4u3).

It should be noted that the traveling ruler in the proof of Theorem 5.1.1 can be gener-

alized for finding 2-cells, i.e., faces in both of a graph embedded on an orientable or

non-orientable surface following.

Generalized Traveling Ruler. Not loss of generality, assume that there are no 2-valent

vertices in G.

(1) Choose an initial vertex v0 of G, a first edge e1 incident with v0 and v1 be the

other end of e1.

(2) The second edge e2 in the boundary walk is the edge after (respective, before) e1

at v1 if e1 is type 0 (respective, type 1). If the edge e1 is a loop, then e2 is the edge after

(respective, before) the other occurrence of e1 at v1.

(3) In general, if the walk traced so far ends with edge ei at vertex vi, then the next

edge ei+1 is the edge after (respective, before) ei at vertex vi if the walk is type 0 (respec-

tive, type 1).

(4) The boundary walk is finished at edge en if the next two edges in the walk would

be e1 and e2 again.

For example, calculation shows that the faces of K 4 embedded on the Klein bottle

shown in fig.5.1.1(b) is

F 1 = (u1, u2, u3, u4, u1), F 2 = (u1, u3, u4, u2, u3, u1, u4, u2, u1).




The general scheme for embedding graphs on locally orientable surfaces was used

extensively by Ringel in the 1950s and then formally proved by Stahl in 1978 following

([Sta1]-[Sta2]).

Theorem 5.1.2 Every rotation system on a graph G defines a unique locally orientable

2-cell embedding of G → S . Conversely, every 2-cell embedding of a graph G → S

defines a rotation system for G.

Proof The proof is the same as that of Theorem 5.1.1 by replacing the traveling ruler

with that of the generalized traveling ruler.

For any embedding of a graph G on a surface S with a band decomposition B, we

can always find a spanning tree T of G such that every edge on this tree is type 0 by the

following algorithm.

Orientability Algorithm. Let T be a spanning tree of G.

(1) Choose a root vertex u for T and an orientation for the 0-band of u0.

(2) For each vertex u1 adjacent to u0 in T , choose the orientation for the 0-band of u1

so that the edge of T from u0 to u1 is type 0.

(3) If ui and ui+1 for an integer are adjacent in T and the orientation at ui has been

already determined but that of ui+1 has not been determined yet, choose an orientation at

ui+1 such that the type of the edge from ui to ui+1 is type 0.

(4) Continuous the process on T until every 0-band has an orientation.

Combining the orientability algorithm with that of Theorem 5.1.2, we get the number

of embeddings of a graph on locally orientable surfaces following.

Corollary 5.1.2 Let G be a connected graph. Then the number of embeddings of G on

locally orientable surfaces is

2 β(G)

v∈V (G)

( ρ(v) − 1)!

and the number of embeddings of G on the non-orientable surfaces is

(2 β(Γ) − 1)

v∈V (Γ)

( ρ(v) − 1)!,

where β(G) = | E (G)| − |V (G)| + 1 is the Betti number of G.

5.1.3 Equivalent Embedding. Two embeddings ( J 1, λ1), ( J 2, λ2) of a graph G on a

locally orientable surface S are called to be equivalent if there exists an orientation-




preserving homeomorphism τ of the surface S such that τ : J 1 → J 2, and τλ = λτ.

If ( J 1, λ1) = ( J 2, λ2) = ( J , λ), then such an orientation-preserving homeomorphism

mapping ( J 1, λ1) to ( J 2, λ2) is called an automorphism of the embedding ( J , λ). Clearly,

all automorphisms of an embedding ( J , λ) form a group under the composition operationof mappings, denoted by Aut( J , λ).

For example, the two embeddings of K 4 shown in Fig.5.1.4(a) and (b) are equivalent,

¹

¹

1

1

2 2

u1u2u3

u4

¹

¹

1

1

2 2u2

u3

u1

u4

(a) (b)

Fig.5.1.4

where the orientation-preserving homeomorphism h is determined by

h(u1) = u1, h(u2) = u3, h(u3) = u2 and h(u4) = u4.

The following result is immediately gotten by definition.

Theorem 5.1.3 Let ( J , λ) be an embedding of a connected graph G on a locally ori-

entable surface S . Then

Aut( J , λ) ≤ AutG.

5.1.4 Euler-Poincare Characteristic. Applying Theorems 4.2.5-4.2.6, we get the Euler-

Poincare characteristic of an embedded graph G on a surface S following.

Theorem 5.1.4 Let G be a graph embedded on a surface S . Then

ν(G) − ε(G) + φ(G) = χ(S ),

where, ν(G), ε(G) and φ(G) are the order, size and the number of faces of the embedded




graph G on S , and χ(S ) is the Euler-Poincar´ e characteristic of S determined by

χ(S ) =

2 i f S ∼ El S 2,

2

−2 p i f S

∼ El T 2#T 2#

· · ·#T 2 p

,

2 − q i f S ∼ El P2#P2# · · · #P2 q

.

§5.2 COMBINATORIAL MAPS

5.2.1 Combinatorial Map. The embedding characteristic of a graph G on surfaces S ,

particularly, Theorems 5.1.1-5.1.2 and the generalized traveling ruler present embryonicmaps. In fact, a map is nothing but a graph cellularly embedded on a surface. That

is why one can enumerates maps by means of embedded graphs on surfaces. In 1973,

Tutte found an algebraic representation for the embedding of graphs on locally orientable

surfaces (see [Tut1]-[Tut2] for details), which completely transfers 2-cell partitions of

surfaces to permutations in algebra.

Let G be an embedded graph on a surface S with a band decomposition B and e ∈ E (G). Then the band Be of e is a topological space B with a homeomorphism h : I × I → B

and sides h(i ×

I ) for i = 0, 1. For characterizing its embedding behavior, i.e., initial and

end vertices, left and right sides of 1-band Be, a natural idea is to introduce quadricells for

e, such as those shown in Fig.5.2.1 following,

.....................................................................u v

Be

u v¹

¹

K xe

xe

α xe

β xe

αβ xe

¹

Fig.5.2.1

where we denote one quarter beginning at the vertex u of Be by xe and its reflective quar-

ters on the symmetric axis e, on the perpendicular mid-line of e and on the central point

of e by α xe, β xe and αβ xe, respectively.

Let K = 1,α,β,αβ. Then K is a 4-element group under the composition operation

by definition with

α2 = 1, β2 = 1, αβ = βα,



Sec.5.2 Combinatorial Maps 175

called the Klein group. The action of K on an edge e ∈ E (G) is defined to be

Ke = xe, α xe, β xe, αβ xe,

called the quadricells of e. Notice that Theorems 5.1.1-5.1.2 and the generalize travelingruler claim the embedded graph G on surface S is correspondent with

ρ L(G) = ( J (v), λ)|J (v) ∈ ρ(G), λ : E (G) → Z2.

Whence, if we turn 1-bands to quadricells for e ∈ E (G), the rotation system (u) at a

vertex u becomes to two cyclic permutations ( xe1, xe2

, · · · , xe ρ(u)), (α xe1

, α xe ρ(u), · · · , α xe2

) if

N G(u) = e1, e2, · · · , e ρ(u). By definition, K xe1∩ K xe2

= ∅ if e1 e2. We therefore get a

set

X α,β = e∈ E (G)

K xe = e∈ E (G)

xe, α xe, β xe, αβ xe.

Define a permutation

P =

u∈V (G)

( xe1, xe2

, · · · , xe ρ(u))(α xe1

, α xe ρ(u), · · · , α xe2

) =

u∈V (G)

C v · (αC −1v α−1),

called the basic permutation on X α,β, i.e., P k x α x for any integer k ≥ 1, x ∈X α,β, where C v = ( xe1

, xe2, · · · , xe ρ(u)

). This permutation also make one understanding

the embedding of G on surface S if we view a vertex u ∈ V (G) as the conjugate cycles

C · (αC −1

α−1

) = ( xe1 , xe2 , · · · , xe ρ(u) )(α xe1 , α xe ρ(u) , · · · , α xe2 ) and an edge e as the quadricellK xe. We have two claims following.

Claim 1. αPα−1 = P−1.

Let P =

u∈V (G)( xe1

, xe2, · · · , xe ρ(u)

)(α xe1, α xe ρ(u)

, · · · , α xe2). Calculation shows that

αPα = α

u∈V (G)

( xe1, xe2

, · · · , xe ρ(u))(α xe1

, α xe ρ(u), · · · , α xe2

)

α−1

= u∈V (G) α( xe1, xe2

,

· · ·, xe ρ(u)

)α−1

· α(α xe1, α xe ρ(u)

,

· · ·, α xe2

)α−1

=

u∈V (G)

(α xe1, α xe2

, · · · , α xe ρ(u))( xe1

, xe ρ(u), · · · , xe2

) = P−1.

Claim 2. The group α,β, P is transitive on X α,β.

For ∀ x, y ∈ X α,β, assume they are the quadricells of edges e1 and e2. By the con-

nectedness of G, we know that there is a path P = e1e2 · · · es connected e′ and e′′ in G for




an integer s ≥ 0. Notice that edges e′ with e1 and e′′ with es are adjacent. Not loss of

generality, let Pk 1 x = xe1 and Pk 2 xes = y. Then we know that

(αβ)

s

xe1

= xes

, or α xes

, or β xes

or αβ xes

.

Whence, we must have that

Pk 2 (αβ)sPk 1 x = y, or Pk 2 α(αβ)sPk 1 x = y, or

Pk 2 β(αβ)sPk 1 x = y, or Pk 2 α(αβ)s+1Pk 1 x = y.

Notice that Pk 2 (αβ)sPk 1 , P k 2 α(αβ)sP k 1 , Pk 2 β(αβ)sPk 1 and Pk 2 α(αβ)s+1P k 1 are ele-

ments in the group α, β, P. Thus α, β, P is transitive on X α,β.

Claims 1 and 2 enable one to define a map M algebraically following.

Definition 5.2.1 Let X be finite set, K = 1,α,β,αβ the Klein group and

X α,β = x∈ X

x, α x, β x, αβ x.

Then a map M is defined to be a pair (X α,β, P) , where P is a basic permutation action

on X α,β such that the following axioms hold:

Axiom 1. αP = P−1α;

Axiom 2. The group Ψ J = α, β, P with J = α, β, P is transitive on X α,β.

Notice that Axiom 2 enables one to decompose P to a production of conjugate

cycles C v and αC −1v α−1 correspondent to the vertices of the M , i.e.,

P =

v∈V ( M )

C v · αC −1v α−1.

We present an example for maps correspondent to embedded graphs following.

Example 5.2.1 The embedded graph K 4 on the tours T 2 shown in Fig.5.2.2 following

can be algebraic represented by a map (X α,β, P) with X α,β = x, y, z, u, v, w, α x, α y, α z,

αu, αv, αw, β x, β y, β z, βu, βv, βw, αβ x, αβ y, αβ z, αβu, αβv, αβw and

P = ( x, y, z)(αβ x, u, w)(αβ z, αβu, v)(αβ y, αβv, αβw)

× (α x, α z, α y)( β x, αw, αu)( β z, αv, βu)( β y, βw, βv).




1

1

2 2

¶

·

×

xy

z

u

v

w

¹

¹

Fig 5.2.2

Its four vertices are

u1 = ( x, y, z), (α x, α z, α y), u2 = (αβ x, u, w), ( β x, αw, αu),

u3 = (αβ z, αβu, v), ( β z, αv, βu), u4 = (αβ y, αβv, αβw), ( β y, βw, βv).

and its six edges are e, αe, βe, αβe, where, e ∈ x, y, z, u, v, w.

5.2.2 Dual Map. Let M = (X α,β, P) be a map. Notice that

αPα−1 = P−1 ⇒ β(Pαβ) β−1 = (Pαβ)−1

and Ψ J = α, β, P is transitive on X β,α also. We known that M ∗ = (X β,α, Pαβ) is also a

map by definition, called the dual map of M . Now the generalized traveling ruler becomes

Traveling Ruler on Map. For ∀ x ∈ X α,β , the successor of x is the element y after αβ x

in P , thus each face of M is a pair of conjugate cycles in the decomposition

Pαβ =

f ∈V ( M ∗)

C ∗ · ( βC −∗ β−1),

i.e., a vertex of its dual map M ∗. The length of a face f of M is called the valency of f .

Example 5.2.2 The faces of K 4 embedded on torus shown in Fig.5.2.2 are respective

f 1 = ( x, u, v, αβw, αβ x, y, αβv, αβ z)( β x, α z, αv, β y, α x, αw, βv, βu),

f 2 = (α y, βw, αu, β z)(αβ y, z, αβu, w).

By the definitions of map M with its dual M ∗, we immediately get the following

results according to Theorems 5.1.1-5.1.2.

Theorem 5.2.1 Every map M = (X α,β, P) defines a unique locally orientable 2-cell

embedding of G → S with

V (G) = C · αC −1α−1 | C ∈ C , E (G) = K x | x ∈ X




and the face set F (G) determined by cycle pairs F , βF β−1 in the decomposition of Pαβ.

Conversely, every 2-cell embedding of a graph G → S defines a map M = (X α,β, P)

determined by

X α,β = e∈ E (G)

K xe = e∈ E (G)

xe, α xe, β xe, αβ xe

and

P =

u∈V (G)

( xe1, xe2

, · · · , xe ρ(u))(α xe1

, α xe ρ(u), · · · , α xe2

),

if N G(u) = e1, e2, · · · , e ρ(u).

By Theorem 5.2.1, the embedded graph G (the map M ) correspondent to the map M

(the embedded graph G) is called the underlying graph of M (map underlying G), denoted

by G( M ) and M (G), respectively.

Theorem 5.2.2 Let M = (X α,β, P) be a map. Then its Euler-Poincar´ e characteristic is

χ( M ) = ν( M ) − ε( M ) + φ( M ),

where ν( M ), ε( M ), φ( M ) are the number of vertices, edges and faces of the map M, re-

spectively.

Example 5.2.2 The Euler-Poincare characteristic χ( M ) of the map shown in Fig.5.2.2 is

χ( M ) = ν( M ) − ε( M ) + φ( M ) = 4 − 6 + 2 = 0.

5.2.3 Orientability. For defining a map (Xα,β, P) is orientable or not, we first prove the

following result.

Theorem 5.2.3 Let M = (X α,β, P) be a map. Then the number of orbits of the group

Ψ L = αβ, P action on X α,β with L = αβ, P is at most 2.

Proof Notice that |Ψ J : Ψ L| = 2, i.e., α, β, P = αβ, P

α αβ, P. For x, y ∈

X , if there are no elements h ∈ Ψl such that xh

= y, by Axiom 2 there must be an elementθ ∈ Ψ J with xθ = y. Clearly, θ ∈ αΨ L. Let θ = αh. Then α xh = y and β x = y, i.e., x, αβ x

in one orbit and α x, β x in another. This fact enables us to know the number of orbits of

Ψ L action on X α,β is 2.

If a map M = (X α,β, P) is on an orientable surface, i.e., each 1-band is type 0, then

any x ∈ X α,β can be not transited to α x by the generalized traveling ruler on its edges,




i.e., the number of orbits of Ψ L action on X α,β is 2. This fact enables us to introduce the

orientability of map following.

Definition 5.2.2 A map M = (X α,β, P) is non-orientable if it satisfies Axiom 3 following,

otherwise, orientable.

Axiom 3. The group Ψ L = αβ, P is transitive on X α,β.

Definition 5.2.3 Let M be a map on a surface S . Then the genus g(S ) is called the genus

of M, i.e.,

g( M ) =

0 i f S ∼ El S 2,

p i f S ∼ El T 2#T 2# · · · #T 2

p

,

q i f S

∼ El P2#P2#

· · ·#P2

q

.

It can be shown that the number of orbits of the group Ψ L action on Xα,β = x, y, z, u, v,

w, α x, α y, α z, αu, αv, αw, β x, β y, β z, βu, βv, βw, αβ x, αβ y, αβ z, αβu, αβv, αβw in Fig.5.2.2

is 2. Whence, it is an orientable map and the genus g( M ) satisfies

2 − 2g( M ) = ν( M ) − ε( M ) + φ( M ) = 4 − 6 + 2 = −2.

Thus g( M ) = 1, i.e., M is on the torus T 2, being the same with its geometrical meaning.

5.2.4 Standard Map. A map M is standard if it only possesses one vertex and one face.

We show that all the standard surfaces in Chapter 4 is standard maps. From Theorem

4.2.4 we have known the standard surface presentations as follows:

(1) The sphere S 2 =

a|aa−1

;

(2) The connected sum of p tori

T 2#T 2# · · · #T 2 p

=

ai, bi, 1 ≤ i ≤ p |

pi=1

aibia−1i b−1

i

;

(3) The connected sum of q projective planes

P2#P2 · · · #P2 q

=

ai, 1 ≤ i ≤ q |

qi=1

ai

.

All of these surface presentations is in fact maps, i.e.,

(1′) The sphere O0 = (X α,β, P) with X α,β(O0) = a, αa, βa, αβa and P(O0) =

(a, αβa)(αa, β);




(2′) The connected sum of p tori O p = (X α,β, P) with

X α,β(O p) =

p

i=1

ai, αai, βai, αβai

p

i=1

bi, αbi, βbi, αβbi

,

P(O p) = (a1, b1, αβa1, αβb1, a2, b2, αβa2, αβb2, · · · , a p, b p, αβa p, αβb p)

(αa1, βb p, βa p, αb p, αa p, · · · , βb2, βa2, αb2, αa2, βb1, βa1, αb1).

(3′) The connected sum of q projective planes N q = (X α,β, P) with

X α,β( N q) =

pi=1

ai, αai, βai, αβai,

P( N q) = (a1, βa1, a2, βa2, · · · , a p, βa p)(αa1, αβa p, αa p, · · · , αβa2, αa2, αβa1).


Theorem 5.2.4 These maps O0, O p and N q are standard maps. Furthermore,

(1) The map O p is orientable with genus g(O p) = p for integers p ≥ 0;

(2) The map N q is non-orientable with genus g( N q) = q for integers q ≥ 1.

Proof Clearly, ν(O p) = 1 and ν( N q) = 1 by definition. Calculation shows that

P(O0)αβ = (a, αβa)(αa, βa);

P(O p)αβ = (a1, αβb1, αβa1, b1, a2, αβb2, αβa2, b2,

· · ·, a p, αβb p, αβa p, b p)

( βa1, βb p, αa p, αb p, βa p, · · · , βb2, αa2, αb2, βa2, βb1, αa1, αb1);

P( N q)αβ = (a1, αa1, a2, αa2, · · · , aq, αaq)( βa1, αβaq, βaq, · · · , αβa2, βa2, αβa1).

Therefore, there only one face in O p and N q. Consequently, they are standard maps for

integers p ≥ 0 and q ≥ 1.

Obviously, the number of orbits of Ψ L action on X α,β(O p) is 2, but that on X α,β(O p) is

1. Whence, O p is orientable for integers p ≥ 0 and N q is non-orientable for integers q ≥ 1.

Calculation shows that the Euler-Poincare characteristics of O p and N q are respective

χ(O p) = 1 − 2 p + 1 and χ( N q) = 1 − q + 2.

Whence, g(O p) = p and g( N q) = q.

By the view of map, the standard surface presentation in Theorem 4 .2.4 is nothing

but the dual maps (X α,β, P) of bouquets B2 p, Bq on T 2#T 2# · · · #T 2 p

or P2#P2# · · · #P2 q



Sec.5.3 Map Groups 181

with

P( B2 p) = (a1, αβb1, αβa1, b1, a2, αβb2, αβa2, b2, · · · , a p, αβb p, αβa p, b p)

( βa1, βb p, αa p, αb p, βa p, · · · , βb2, αa2, αb2, βa2, βb1, αa1, αb1);P( Bq) = (a1, αa1, a2, αa2, · · · , aq, αaq)( βa1, αβaq, βaq, · · · , αβa2, βa2, αβa1).

For example, we have shown this dual relation in Fig.5.2.3 for p = 1 and q = 2

following.

O O

¹

¹

a

b

αβa

αβb

¹

a

βab

βb

¹

¹

a

αβbαβa

b a

αab

αb

Fig.5.2.3

In fact, the embedded graph B2 on torus and Klein bottle are maps (X α,β, P), where

X α,β( B2) = a, αa, βa, αβa, b, αb, βb, αβb, P = (a, αβb, αβa, b)(αa, αb, βa, βb), Pαβ =

(a, b, αβa, αβb)(αa, βb, βa, αb) on the torus, and P = (a, αa, b, αb)( βa, αβb, βb, αβa),

Pαβ = (a, βa, b, βb)(αa, αβb, αb, αβa) on the Klein bottle, respectively.

§5.3 MAP GROUPS

5.3.1 Isomorphism of Maps. Let M 1 = (X 1α,β, P1) and M 2 = (X 2

α,β, P2) be maps. If

there exists a bijection

ξ : X 1α,β → X 2

α,β

such that for ∀ x ∈ X 1

α,β,

ξα( x) = αξ ( x), ξβ( x) = βξ ( x) and ξ P1( x) = P2 ξ ( x).

Such a bijection ξ is called an isomorphism from maps M 1 to M 2.

Clearly, ξ −1α( y) = αξ −1( y), ξ −1 β( y) = βξ −1( y) and ξ −1P( y) = P ξ −1( y) for y ∈X 2

α,β. Thus the bijection ξ −1 : X 2α,β

→ X 1α,β is an isomorphism from maps M 2 to M 1.




Whence, we can just say such M 1 and M 2 are isomorphic without distinguishing that the

isomorphism ξ is from M 1 to M 2 or from M 2 to M 1 if necessary.

Theorem 5.3.1 Let M 1 and M 2 be isomorphic maps. Then

(1) M 1 is orientable if and only if M 2 is orientable;

(2) ν( M 1) = ν( M 2) , ε( M 1) = ε( M 2) and φ( M 1) = φ( M 2) , particularly, the Euler-

Poincar´ e characteristics χ( M 1) = χ( M 2).

Proof Let M 1 = (X 1α,β

, P1), M 2 = (X 2α,β

, P2), τ : X 1α,β

→ X 2α,β an isomorphism

from M 1 to M 2 and x1, x2 ∈ X 1α,β such that there exists a σ ∈ Ψ1

L = αβ, P1 with

σ( x1) = x2. Then There must be τστ−1(τ( x1)) = τ( x2), i.e., τΨ1 L

τ−1 = αβ, P2 = Ψ2 L.

Whence, Ψ1 L is not transitive on X 1

α,β if and only if Ψ2 L is not transitive on X 2

α,β. That is

the conclusion (1).For (2), let x1 be an element in the conjugate pair C · (αC −1α−1) of P1 and y1 an

element in C ′ · (αC ′−1α−1) of P2. It is easily know that τ(C · (αC −1α−1)) = C ′ · (αC

′−1α−1)

and τ( x1, α x1, β x1, αβ x1) = y1, α y1, β y1, αβ y1, i.e., τ : K x1 → Ky1. Whence, τ is

an bijection between V ( M 1) and V ( M 2), E ( M 1) and E ( M 2). Thus ν( M 1) = ν( M 2) and

ε( M 1) = ε( M 2).

By definition, we know that τ(P1αβ) = (P2αβ)τ. So similarly we know that τ is

also a bijection between the vertices, i.e., faces of M 1 and M 2. Consequently, we get that

φ( M 1) = φ( M 2).

For ∀ x ∈ X α,β, let v x, e x and f x be the vertex, edge and face containing the quadricell

x in a map M = (X α,β, P). The triple (v x, e x, f x) is called a flag incident with that of x in

M . Denoted by F ( M ) all flags in a map M . Then we get the following result by the proof

of Theorem 5.3.1.

Corollary 5.3.1 Let M 1 and M 2 be isomorphic maps. Then there is a bijection between

flag sets F ( M 1) and F ( M 2).

Theorem 5.3.2 A map M 1 = (X 1

α,β, P1) is isomorphic to M 2 = (X 2

α,β, P2) if and only if the dual map M ∗

1 = (X 1 β,α

, P1αβ) is isomorphic to that of M ∗2 = (X 2

β,α, P2αβ).

Proof Let τ : X 1α,β

→ X 2α,β be an isomorphism from M 1 to M 2. Then τα − ατ,

τβ = βτ and τP1 = P2τ. Consequently, τ(P1αβ) = P2τ(αβ) = (P2αβ)τ. Notice that

X 1α,β = X 1

β,α and X 2α,β = X 2

β,α. We therefore know that τ is an isomorphism between M ∗1and M ∗

2.




Applying isomorphisms between maps, an alternative approach for determining equiv-

alent embeddings and maps on locally orientable surfaces underlying a graph can be de-

fined as follows:

For a given map M underlying a graph G, it is obvious that Aut M |G ≤ Aut 12

G.

Whence, we can extend the action of ∀g ∈ Aut 12

G on V (G) to that of g| 12 on Xα,β with

X = E (G) by defining that for ∀ x ∈ Xα,β, if xg = y, then

xg| 12

= y, (α x)g| 12

= α y, ( β x)g| 12

= β y and (αβ x)g| 12

= αβ y.

Then we can characterize equivalent embeddings and isomorphic maps following.

Theorem 5.3.3 Let M 1 = (X α,β, P1) and M 2 = (X α,β, P2) be maps underlying a graph

G. Then

(1) M 1 and M 2 are equivalent if and only if there is an element ζ ∈ Aut 12

G such that

P ζ

1 = P2.

(2) M 1 and M 2 are isomorphic if and only if there is an element ζ ∈ Aut 12

G such that

P ζ

1 = P2 or Pζ

1 = P−12 .

Proof Let κ be an equivalence between embeddings M 1 and M 2. Then by definition,

κ must be an isomorphism between maps M 1 and M 2 induced by an automorphism ι ∈AutG. Notice that

AutG AutG| 12 ≤ Aut 1

2G.

We know that ι ∈ Aut 12

G.

Now if there is a ζ ∈ Aut 12

G such that Pζ

1= P2, then ∀e x ∈ X 1

2(G), ζ (e x) = ζ (e)ζ ( x).

Assume that e = ( x, y) ∈ E (G), then by convention, we know that if e x = e ∈ X α,β, there

must be e y = βe. Now by the definition of automorphism on the semi-arc set X 12

(G), if

ζ (e x) = f u, where f = (u, v), then there must be ζ (e y) = f v. Notice that X 12

(G) = X β. We

therefore know that ζ (e y) = ζ ( βe) = β f = f v. Now extend the action of ζ on X 12

(G) to

X α,β by ζ (αe) = αζ (e). We get that ∀e ∈ X α,β,

αζ (e) = ζα(e), βζ (e) = ζβ(e) and Pζ

1 (e) = P2(e).

So the extend action of ζ on X α,β is an isomorphism between the map M 1 and M 2, which

preserve the orientation on M 1 and M 2. Whence, ζ is an equivalence between the map M 1

and M 2. That is the assertion (1).




For the assertion (2), if there is an element ζ ∈ Aut 12

G such that P ζ

1= P2, then the

map M 1 is isomorphic to M 2. If Pζ

1 = P−12 , then there must be P ζα

1 = P2. So M 1 is also

isomorphic to M 2. This is the sufficiency of (2).

Let ξ be an isomorphism between maps M 1 and M 2. Then for ∀ x ∈ X α,β,

αξ ( x) = ξα( x), βξ ( x) = ξβ( x) and P ξ

1 ( x) = P2( x).

By convention, the condition

βξ ( x) = ξβ( x) and P ξ

1 ( x) = P2( x)

is just the condition of an automorphism ξ or αξ on X 12

(G). Whence, the assertion (2) is

also true.

5.3.2 Automorphism of Map. If M 1 = M 2 = M , such an isomorphism between M 1 and

M 2 is called an automorphism of M , which surveys symmetries on a map.

Example 5.3.1 Let M = (X α,β, P) be a map with

X α,β( B2) = a, αa, βa, αβa, b, αb, βb, αβb

and

P = (a, αβb, αβa, b)(αa, αb, βa, βb),

i.e., the bouquet B2 on the torus shown in Fig.5.3.1 following.

O

¹

¹

¹

a

αβbαβa

b

Fig.5.3.1

We determine its automorphisms following. Define

τ1 =

a αa βa αβa b αb βb αβb

αa a αβa βa βb αβb b αb

= (a, αa)( βa, αβa)(b, βb)(αb, αβb),




τ2 =


βa αβa a αa αb b αβb βb

= (a, βa)(αa, αβa)(b, αb)( βb, αβb),

τ3 = a αa βa αβa b αb βb αβb

αβa βa αa a αβb βb αb b

= (a, αβa)(αa, βa)(b, αβb)(αb, βb),

τ4 =


b αb βb αβb αβa βa αa a

= (a, b, αβa, αβb)(αa, αb, βa, βb),

τ5 =


αb b αβb βb αa a αβa βa

= (a, αb)(αa, b)( βa, αβb)(αβa, βb),

τ6 =


βb αβb b αb βa αβa a αa

= (a, βb)(αa, αβb)( βa, b)(αβa, αb),

τ7 =


αβb βb αb b a αa βa αβa

= (a, αβb, αβa, b)(αa, βb, βa, αb).

We are easily to verify that these permutations 1X α,β, τi, 1

≤i

≤7 are automorphisms of

the map M shown in Fig.5.3.1.

Theorem 5.3.4 All automorphisms of a map M = (X α,β, P) form a group.

Proof Let τ, τ1 and τ2 be automorphisms of M . Then we know that τα = ατ, τβ =

βτ, τP = Pτ and τ1α = ατ1, τ1 β = βτ1, τ1P = Pτ1. Clearly, 1X α,βis an automorphism

of M and τ−1α = ατ−1, τ−1 β = βτ−1, τ−1P = Pτ−1, i.e., τ−1 is an automorphism of M .

Furthermore, it is easily to know that

(ττ1)α = α(ττ1), (ττ1) β = β(ττ1) and (ττ1)P = P(ττ1),

i.e., ττ1 is also an automorphism of M with

x(ττ1)τ2 = xτ(τ1 τ2)

for ∀ x ∈ X α,β, i.e., (ττ1)τ2 = τ(τ1τ2). So all automorphisms form a group by definition.




Such a group formed by all automorphisms of a map M is called the automorphism

group of M , denoted by Aut M and any subgroup Γ of automorphism groups of maps is

called a map group.

Theorem 5.3.5 Any map group Γ is fixed-free.

Proof Let M = (X α,β, P) be a map, x ∈ X α,β and Γ ≤ Aut M . If xσ = x, we prove

that

σ = 1X α,β.

In fact, for ∀ y ∈ X α,β, by definition Ψ J = α, β, P is transitive on X α,β, there exists an

element h ∈ Ψ J such that xh = y. Hence,

yσ = xσh = xhσ = xh = y,

i.e., σ fixes all elements in X α,β.

For a group (Γ; ), denoted by Z Γ( H ) = g ∈ Γ| g h g−1 = h, ∀h ∈ H the

centralizer of H in (Γ; ) for H ≤ Γ. Then we are easily to get the following result for

automorphism group of map.

Theorem 5.3.6 Let M = (X α,β, P) be a map. Then Aut M = Z SX α,β(α, β, P) , where

S X α,βis the symmetric group on X α,β.

Proof Let ∀τ ∈ Aut M be an automorphism. Then we know that τα = ατ, τβ = βτand τP = Pτ by definition. Whence, τ ∈ Z SX α,β

(α,β, P). Conversely, for σ ∈ Z SX α,β

(α, β, P), It is clear that σα = ασ, σβ = βσ and σP = Pσ by definition.

A characterizing for automorphism group of map can be found in the following.

Theorem 5.3.7 Let M = (X α,β, P) be a map with A = Aut M and v ∈ V ( M ). Then the

stabilizer Av is isomorphic to a subgroup H ≤

C v

generated by C v = C v · αC −1v α−1 , i.e.,

a product of conjugate pair of cycles in P .

Proof By Theorem 2.1.1, if g

∈Av, we know that gC vg−1 = C g(v) = C v. That is

gC v = C vg. Whence, if w is a quadricell in C v, then g(w) is also so. Denote the constraint

action of an automorphism g ∈ Av on elements in C v by g. Notice that C v is a product of

conjugate pairs of cycles in P. There must be an integer i such that g(w) = C i

v. Choose

x = C j

v(w) be a quadricell in C v. Then

g( x) = gC i

v(w) = C i+ j

v (w) = C i

v( x).




Whence, g = C i

v. Define a homomorphism θ : Av →

C v

by θ (a) = g for ∀g ∈ Av.

Then it is also a monomorphism by Theorem 5.3.5. Thus Av is isomorphic to a subgroup

H ≤

C v.

Applying isomorphisms between maps, similar to that of Theorem 5.3.3 we can also

characterize automorphisms of a map by extended actions of semi-arc automorphisms of

its underlying graph following.

Theorem 5.3.8 Let M = (X α,β, P) be a map underlying graph G, g ∈ Aut 12

G. Then the

extend action g| 12 of g on X α,β with X = E (G) is an automorphism of map M if and only if

∀v ∈ V ( M ) , g| 12 preserves the cyclic order of v.

Proof Let g| 12 ∈ Aut M be extended by g ∈ Aut 1

2G with ug = v for u, v ∈ V ( M ). Let

u = ( x1, x2, · · · , x ρ(u))(α x ρ(u), · · · , α x2, α x1),

v = ( y1, y2, · · · , y ρ(v))(α y ρ(v), · · · , α y2, α y1).

Then there must be

( x1, x2, · · · , x ρ(u))g| 1

2= ( y1, y2, · · · , y ρ(v)) or

( x1, x2, · · · , x ρ(u))g| 1

2= (α y ρ(v), · · · , α y2, α y1).

Without loss of generality, we assume that ( x1, x2, · · · , x ρ(u))g

|12

= ( y1, y2, · · · , y ρ(v)). Thus,

(g| 12 ( x1), g| 1

2 ( x2), · · · , g| 12 ( x ρ(u))) = ( y1, y2, · · · , y ρ(v)).

Whence, g| 12 preserves the cyclic order of vertices in the map M .

Conversely, if the extend action g| 12 of g ∈ Aut 1

2G on Xα,β preserves the cyclic order

of each vertex in M , i.e., ∀u ∈ V (G), ∃v ∈ V (G) such that ug| 12

= v. Let

P =

u

∈V ( M )

u.

Then

P g| 12

=

u∈V ( M )

ug| 12

=

v∈V ( M )

v = P .

Whence, the extend action g| 12 is an automorphism of map M .

Combining Corollary 5.3.1 and Theorem 5.3.5 enables us to get the following result.




Theorem 5.3.9 Let M = (X α,β, β) be a map with νi of vertices and φi faces of valency

i, i ≥ 1. Then

|Aut M | | (2iνi, 2 jφ j ; i ≥ 1, j ≥ 1),

where (2iνi, 2 jφ j ; i ≥ 1, j ≥ 1) denotes the greatest common divisor of 2iνi, 2 jφ j for an

integer pair i, j ≥ 1.

Proof Let Λi and ∆ j respectively be the sets of quadricells incident with a vertex of

valency i or incident with a face of valency j for integers i, j ≥ 1. Consider the action

of Aut M on Λi and ∆ j. By Corollary 5.3.1, such an action is closed in Λi or ∆ j. Then

applying Theorem 2.1.1(3), we know that

|Aut M | = |(Aut M ) x|| xAut M | = | xAut M |

for ∀ x ∈ Λi for |(Aut M ) x| = 1 by Theorem 5.3.5. Therefore, the length of each orbit of

Aut M action on Λi or ∆ j is the same |Aut M |. Notice that |Λi| = 2iνi and |∆ j| = 2 jφ j. We

get that

|Aut M | | |Λi| = 2iνi and |Aut M | | |∆ j| = 2 jφ j

for any integer pairs i, j ≥ 1. Thus

|Aut M | | (2iνi, 2 jφ j ; i ≥ 1, j ≥ 1).

Corollary 5.3.2 Let M =

(X α,β, P) be a map with vertex valency k and face valency l.Then |Aut M | | (2k | M |, 2l| M ∗|) , where M ∗ is the dual of M. Particularly, |AutO p| | 2 p and

|AutO p| | 2 p for standard maps O p and N q.

By Theorem 5.3.9, we can get automorphism groups Aut M of map M in sometimes.

Example 5.3.2 Let M = (X α,β, P) be the map shown in Fig.5.2.2, i.e., K 4 on torus with

one face length 4 and another 8. By Theorem 5.3.9, there must be |Aut M | | (4×3, 8, 4) = 4,

i.e., |Aut M | ≤ 4. Define

σ1 = ( x, α x)( β x, αβ x)( y, α z)(α y, z)( β z, αβ z)(αβ z, β y)(v, βv)(αv, αβv)(u, αw)(αu, w)( βu, αβw)(αβu, βw)

and

σ2 = ( x, β x)(α x, αβ x)( y, αw)(α y, w)( β y, αβw)(αβ y, βw)

(v, αv)( βv, αβb)( z, αu)(α z, u)( β z, αβu)(αβ z, βu).




It can be verifies that σ1 and σ2 both are automorphisms of M and σ21 == 1X α,β

and

σ22 = 1X α,β

. So Aut M = σ1, σ2.

Example 5.3.3 We have construct automorphisms 1X α,βand τi, 1

≤i

≤7 for the map

shown in Fig.5.3.1 in Example 5.3.1. Consequently, we get that

Aut M = 1X α,β, τ1, τ2, τ3, τ4, τ5, τ6, τ7

by Corollary 5.3.2.

Notice that

2i≥1

iνi = 2i≥1

iφi = |X α,β|

for a map M = (X α,β, P). Therefore, we get the following conclusion.

Corollary 5.3.3 For any map M = (X α,β, P) , |Aut M | | |X α,β| = 4ε( M ).

Proof Applying Theorem 5.3.9, we know that

|Aut M | |i≥1

2iνi and |Aut M | |i≥1

2iφi.

Because of

2

i≥1

iνi = 2

i≥1

iφi = |X α,β|,

we immediately get that |Aut M | | |X α,β| = 4ε( M ).

Now we determine automorphisms of standard maps on surfaces.

Theorem 5.3.10 Let O p = (X α,β(O p), P(O p)) be an orientable standard map with

X α,β(O p) =

pi=1

ai, αai, βai, αβai p

i=1

bi, αbi, βbi, αβbi ,

P(O p) = (a1, b1, αβa1, αβb1, a2, b2, αβa2, αβb2, · · · , a p, b p, αβa p, αβb p)

(αa1, βb p, βa p, αb p, αa p, · · · , βb2, βa2, αb2, αa2, βb1, βa1, αb1).

and let N q = (X α,β, P) be a non-orientable map with

X α,β( N q) =

pi=1

ai, αai, βai, αβai,

P( N q) = (a1, βa1, a2, βa2, · · · , a p, βa p)(αa1, αβa p, αa p, · · · , αβa2, αa2, αβa1).




Define

τs = P4s(O p), 0 ≤ s ≤ p − 1,

σ =

p

i=1

(ai, αai)(bi, βbi)(αβai, βai)(αβbi, αbi),

θ =

pi=1

(ai, αβbi)(αai, βbi), ς =

pi=1

(ai, αβai)(bi, αβbi)

and

ηl = P2l( N q), 0 ≤ l ≤ q − 1; ϑ =

qi=1

(ai, αβai)(αai, βai).

Then

AutO p =

θ, σ, ς, τs, 1

≤s

≤p

−1

and Aut N q

≥ ϑ, ηl, 1

≤l

≤q

−1

.

Proof It is easily to verify that xα = α x, x β = β x, xP(O p) = P(O p) x if

x ∈ θ , σ, ς , τs, 1 ≤ s ≤ p − 1 and yα = α y, y β = β y, yP( N q) = P( N q) y if

y ∈ ϑ, ηl, 1 ≤ l ≤ q − 1. Thus AutO p ≥ θ, σ, ς, τs, 1 ≤ s ≤ p − 1 and Aut N q ≥ϑ, ηl, 1 ≤ l ≤ q − 1. Notice that | θ, σ, ς, τs, 1 ≤ s ≤ p − 1 | = 8 p = |X α,β(O p)|. Ap-

plying Corollary 5.3.3, AutO p = θ, σ, ς, τs, 1 ≤ s ≤ p − 1 is followed.

5.3.3 Combinatorial Model of Klein Surface. For a complex algebraic curve, a very

important problem is to determine its birational automorphisms. For curve C of genus

g ≥ 2, Schwarz proved that Aut(C) is finite in 1879 and then Hurwitz proved | Aut (C)| ≤84(g − 1), seeing [FaK1] for details. As observed by Riemann, the groups of birational

automorphisms of complex algebraic curves are the same as the automorphism groups of

compact Riemann surfaces which can be combinatorially dealt with the approach of maps

on surfaces. Jones and Singerman proved the following result in [JoS1].

Theorem 5.3.11 If M is an orientable map of genus p, then Aut M is isomorphic to a

group of conformal transformations of a Riemann surface.

Notice that the automorphism group of Klein surface possesses the same represen-

tation as that of Riemann surface by Theorem 4.5.7. This enables us to get a result likely

for Klein surfaces following.

Theorem 5.3.12 If M is a locally orientable map on a Klein surface S , then Aut M

is isomorphic to a group of conformal transformations of a Klein surface, particularly,

Aut M ≤ AutS .



Sec.5.4 Regular Maps 191

Proof According to Theorem 4.5.7, there exists a NEC group Γ such that AutS ≃ N Ω(Γ)/Γ, where Ω = Aut H = PGL(2,R) being the automorphism group of the upper half

plane H . Because M is embeddable on Klein surface S , so there is a fundamental region

F , a polygon in H such that gF |g ∈ Γ is a tessellation of H , i.e., S is homeomorphic to H /Γ. By Constructions 4.4.1-4.4.2, we therefore know that Aut M ≤ N Ω(Γ)/Γ, i.e., Aut M

is a subgroup of conformal transformation of Klein surface S .

§5.4 REGULAR MAPS

5.4.1 Regular Map. A regular map M = (X α,β, P) is such a map that its automorphism

group Aut M is transitive on X α,β, i.e., |Aut M |=

4ε( M ). For example, the map discussedin Example 5.3.2 is such a regular map, but that map in Example 5.3.1 is not.

If M is regular, then Aut M is transitive on vertices, edges and faces of M by Corollary

5.3.1. This fact enables us to get the following result.

Theorem 5.4.1 Let M be a regular map with vertex valency k ≥ 3 and face valency l ≥ 3 ,

called a type (k , l) regular maps. Then k ν( M ) = lφ( M ) = 2ε( M ) and

g( M ) =

1 +

(k − 2)(l − 2) − 4

4l

ν( M ), i f M is orientable;

2 + (k

−2)(l

−2)

−4

2l ν( M ), if M is non − orientable.

Proof Let νk = ν( M ), φl = φ( M ) and νi = φ j = 0 if i k , j l in the equalities

2i≥1

iνi = 2i≥1

iφi = |X α,β| = 4ε( M ),

we immediately get that k ν( M ) = lφ( M ) = 2ε( M ).

Substitute ε( M ) =k

2ν( M ) and φ( M ) =

k

lν( M ) in the Euler-Poincare genus formulae

g( M ) = 2 + ε( M ) − ν( M ) − φ( M )

2, if M is orientable

2 + ε( M ) − ν( M ) − φ( M ), if M is non − orientable.

We get that

g( M ) =

1 +

(k − 2)(l − 2) − 4

4l

ν( M ), if M is orientable;

2 +

(k − 2)(l − 2) − 4

2l

ν( M ), if M is non − orientable.




This theorem enables us to find type (k , l) regular maps on orientable or non-orientable

surfaces with small genus following.

Corollary 5.4.1 A map M is regular of g( M ) = 0 if and only if G( M ) = C l, l

≥1 or the

1-skeleton of the five Platonic solids.

Proof If k = 2 then ν( M ) = ε( M ) = l and φ( M ) = 2. Whence, M is a map underlying

a circuit C l on the sphere. Indeed, such a map M is regular by the fact Aut M = ρ,α,

where ρ is the rotation about the center of C l through angles 2π/l from a chosen vertex

u0 ∈ V (C l) with ρl = 1X α,β.

Let k ≥ 3. Then by Theorem 5.4.1, we get that

1 + (k − 2)(l − 2) − 4

4l ν( M ) = 0, i.e., (k

−2)(l

−2) < 4

by Theorem 5.4.1, i.e., (k , l) = (3, 3), (3, 4), (3, 5), (4, 3), (5, 3), which are just the Pla-

tonic solids shown in Fig.5.4.1 following.

tetrahedron hexahedron octahedron

dodecahedron icosahedron

(3,3) (3,4) (4,3)

(3,5) (5,3)

Fig.5.4.1

Corollary 5.4.2 There are infinite regular maps M of torus T 2

.

Proof In this case, we get (k − 2)(l − 2) = 4 by Theorem 5.4.1. Whence, (k , l) =

(3, 6), (4, 4), (6, 3). Indeed, there exist regular maps on torus for such integer pairs. For

regular map on torus with (3, 6) or (4, 4), see (a) or (b) in Fig.5.4.2. It should be noted

that the regular map on torus with (6, 3) is just the dual that of (3, 6) and we can construct

such regular maps of order 6s or 4s for integer s ≥ 1. So there are infinite many such




regular maps on torus.

¹

¹

1 2 3 4 5

1 2 3 4 5

1’ 1’

2’ 2’

1 2

1 2

1’ 1’

2’ 2’

(a) (b)

¹

¹

Fig.5.4.2

Corollary 5.4.3 There are finite regular maps on projective plane P2 with vertex valency≥

3 and face valency≥ 3.

Proof Similarly, we know that (k − 2)(l − 2) < 4 by Theorem 5.4.1, i.e., the possible

types of M are (3, 3), (3, 4), (4, 3), (5, 3), (5, 3) and it can verified easily that there are no

(3, 3) regular maps on P2. Calculation shows that

(k , l) ν( M ) ε( M ) G( M ) Existing? M Existing?

(3, 3) 2 3 Yes No

(3, 4) 4 6 Yes Yes

(4, 3) 3 6 Yes Yes

(3, 5) 10 15 Yes Yes

(5, 3) 6 15 Yes Yes

Therefore, regular maps on projective plane P2 with vertex valency≥ 3 and face valency≥3 is finite. The regular maps of types ((3, 5)) and (3, 4) are shown in Fig.5.4.3.

1

1

2

2

3 3

5

5

4

4

1 2

2 1

(a) (b)

Fig.5.4.3




The following result approves the existence of regular maps on every orientable sur-

face.

Theorem 5.4.2 For any integer p

≥0 , there are regular maps on every orientable surface

of genus p.

Proof Applying Theorem 5.3.10, the standard map O p is regular on the orientable

surface of genus p. Combining the result in Corollary 5.4.1, we get the conclusion.

Notice that Theorem 4.5.2 has claimed that the automorphism group of a Klein sur-

face is finite. In fact, by Theorem 5.4.1, we can also determine the upper bound of Aut M

for regular maps M on a surface of genus g ≥ 2.

Theorem 5.4.3 Let M be a regular map on a surface S of genus g ≥ 2 with vertex valency

k ≥ 3 and face valency l ≥ 3. Then

|Aut M | ≤ 168(g − 1), i f S is orientable,

84(g − 1), i f S is non − orientable.

and with the equality holds if and only if (k , l) = (3, 7) or (7, 3).

Proof By definition, a map M = (X α,β, P) on S is regular if and only if |Aut M | =

|X α,β| = 4ε( M ). Substitute ν( M ) =2

k ε( M ) in Theorem 5.4.1, we get that

|Aut M | = 8kl

(k − 2)(l − 2) − 4 (g−

1), if S is orientable,4kl

(k − 2)(l − 2) − 4

(g − 1), if S is non − orientable.

Clearly, the maximum value of kl

(k − 2)(l − 2) − 4is 21 occurring precisely at (k , l) = (3, 7)

or (7, 3). Therefore,

|Aut M | ≤

168(g − 1), i f S is orientable,

84(g − 1), i f S is non − orientable.

and with the equality holds if and only if (k , l) = (3, 7) or (7, 3).

5.4.2 Map NEC-Group. We have known that Ψ J = α, β, P acts transitively on X α,β,

i.e., xΨ J = X α,β. Furthermore, if M is regular, then its vertex valency and face valency

both are constant, say n and m. Usually, such a regular map M is called with type (n, m).

Then we get the presentation of Ψ J for M following

Ψ J =

α, β, P | α2 = β2 = Pn = (Pαβ)m = 1X α,β

.




We regard relations of the form P∞ = 1X α,βor (Pαβ)∞ = 1X α,β

as vacuous. The free

group Ψ generated by α,β, P, i.e., Ψ = α, β, P is called the universal map of M , a

tessellation of planar Klein surface H . It should be note that Ψ J is isomorphic to the NEC

group generated by facial boundaries of M . Whence, M ≃ H / xΨ

J = xΨ/ xΨ

J ≃ Ψ/Ψ J ,where x is a chosen point in H . Applying Theorem 4.5.9, we get the following result.

Theorem 5.4.4 Let M = (X α,β) be a regular map on a Klein surface S . Then Aut M ≃ N Ψ(Ψ J )/Ψ J , where N Ψ(Ψ J ) is the normalizer of Ψ J in Ψ.

This result will be applied for constructing regular maps on surfaces in Section 5 .5.

5.4.3 Cayley Map. Let (Γ; ) be a finite group generated by S . A Cayley map of Γ to S

with 1Γ S and S −1 = S , denoted by CayM(Γ : S , r ) is a map (X α,β(Γ : S ), P(Γ : S )),

where

X α,β(Γ : S , r ) = gh, αgh, βgh, αβgh | g ∈ Γ, h ∈ S and g−1 h ∈ S ,

P(Γ : S , r ) =

g∈Γ, h∈S

(gh, gr (h), gr 2(h), · · · · · ·)(αgh, αgr −1(h), αgr −2(h), · · · · · ·)

with ταgh = ατgh, τβgh = βτgh for τ ∈ Γ, where r : S → S is a cyclic permutation.

Clearly, the underlying graph of a Cayley map CayM(Γ : S , r ) is Cay(Γ : S ).

Example 5.4.1 Let (Γ; ) be the Klein group Γ = 1,α,β,αβ, S = α,β,αβ and r =

(α,β,αβ). Then the Cayley map CayM

(Γ : S , r ) is K 4 on the plane shown in Fig.5.4.4.

α

1

βαβ

β αβ

α

α

βαβ

Fig.5.4.4

Theorem 5.4.5 Any Cayley map CayM(Γ : S , r ) is vertex-transitive. In fact, there is a

regular subgroup of AutCayM(Γ : S , r ) isomorphic to Γ.

Proof Consider the action of left multiplication LΓ on vertices of CayM(Γ : S , r ),




i.e., Lσ : h → g h for g, h ∈ Γ. We have known it is transitive on vertices of Cayley

graph Cay(Γ : S ) by Theorem 3.2.1. It only remains to show that such a permutation

Lg is a map automorphism of CayM(Γ : S , r ). In fact, for gh ∈ X α,β(Γ : S , r ) we know

Lσαgh = σαgh = ασgh = α Lσgh i.e., Lσα = α Lσ by definition. Similarly, Lσ β = β Lσ.Notice that if g−1 h ∈ S , then (σ g)−1 (σ h) = g−1 h ∈ S , i.e., ( Lσ(g)) Lσ(h) ∈

X α,β(Γ : S , r ). Calculation shows that

LσP(Γ : S , r ) L−1σ

= Lσ

g∈Γ, g−1h∈S

(gh, gr (h),gr 2(h)

, · · ·)(αgh, αgr −1(h),αgr −2(h)

, · · ·) L−1σ

=

g∈Γ, g−1h∈S

( Lσ(g) Lσ(h), Lσ(g) Lσ(r (h)), · · ·)(α Lσ(g) Lσ(h), α Lσ(g) Lσ(r −1(h)), · · ·)

= g∈Γ, g−1h∈S

(σgσh, σgσr (h), σgσr 2(h), · · ·)(ασgσh, ασg)σr −1(h), ασgr −2(σh), · · ·)

=

s∈Γ, s−1t ∈S

(st , sr (t ), sr 2(t ), · · ·)(αst , αsr −1(t ),αsr −2 (t )

, · · ·) = P(Γ : S ),

i.e., Lg is an automorphism of CayM(Γ : S , r ). We have known that LΓ ≃ Γ by Theorem

1.2.14.

Although every Cayley map is vertex-transitive, there are non-regular Cayley maps

on surfaces. For example, let (Γ; ) be an Abelian group with Γ = 1Γ, a, b, c, S = a, b, c,

a2 = b2 = c2 = 1Γ, a

b = b

a = c, a

c = c

a = b, b

c = c

b = a and r = (a, b, c).

Then the Cayley map CayM(Γ : S , r ) is K 4 on the projective plane shown in Fig.5.4.5,

which is not regular.

¹

a

1’

a

a

a

1Γ

1’

b

b

c

c

cb

Fig.5.4.5

Now we find regular maps in Cayley maps of finite groups. First, we need to prove

the following result.






Proof If σ(u) = v, σ(v) = v for two vertices u, v ∈ V ( M ), let uv = x, α x, β x, αβ x,

then there must be σ( x) = x because of uv ∈ V ( M ). Applying Theorem 5.3.5, we get the

conclusion.

A Frobenius group Γ is defined to be a transitive group action on a set Ω such that

only 1Γ has more than one fixed points in Ω. By Theorem 5.4.8, thus the automorphism

group Aut M of a complete vertex-transitive map M is necessarily Frobenius. For finding

complete regular map, we need a characterization due to Frobenius in 1902 following.

Theorem 5.4.9 Let Γ be a Frobenius group action on Ω with N ∗ the set of fixed-free

elements of Γ and N = N ∗ ∪ 1Γ. Then there are must be

(1) | N | = |Ω|;(2) N is a regular normal subgroup of Γ.

Theorem 5.4.10 Let Γ be a sharply 2-transitive group action on Ω. Then |Ω| is a prime

power.

A complete proof of Theorems 5.4.9 and 5.4.10 can be found in [Rob1] by applying

the character theory on linear representations of groups. But if the condition that Γ x is

Abelian for a point x ∈ Ω is added, Theorem 5.4.9 can be proved without characters of

groups. See [BiW1] for details.

Theorem 5.4.11 Let M be a complete map. Then Aut M acts transitively on the vertices

of M if and only if M is a Cayley map.

Proof The sufficiency is implied in Theorem 5.4.5. For the necessity, applying The-

orem 5.4.8 we know that Aut M is a Frobenius group. Now by Theorem 5.3.7, (Aut M ) x

is isomorphic to a subgroup generated by C v = C v · αC −1v α−1, i.e., a product of conjugate

pair of cycles in P. Whence, we get a regular normal subgroup N of Aut M by Theorem

5.4.9. Let Γ = Zn and define a bijection σ : V (Cay M (Zn,Zn \ 1, r )) → N by σ(i) = ai,

where ai is the unique element transforming point 0 to i in N . Calculation shows thatr : N \ 1 → N \ 1 is given by r (ai) = aP(Zn ,Zn\1,r ))(i) for i 0. Thus we get a Cay-

ley map Cay M (Zn,Zn \ 1, r ). It can be verified that the bijection σ is an automorphism

between maps M and Cay M (Zn,Zn \ 1, r ).

Now we summarize all properties of Aut M in the following obtained in previous on

regular map M underlying K n:



Sec.5.5 Constructing Regular Maps by Groups 199

(1) Aut M is a Frobenius group of order n(n − 1);

(2) Aut M has a regular normal subgroup isomorphic to Zm p for a prime p and an

integer m ≥ 1, i.e., n = pm;

(3) Aut M is transitive on vertices, edges and faces of M , and regular on X α,β;

(4) For ∀v ∈ V ( M ), (Aut M )v ≃ Zn−1.

We prove the main result on complete regular maps of this subsection following.

Theorem 5.4.12 A complete map M underlying K n is regular on an orientable surface if

and only if n is a prime power.

Proof If M is regular on an orientable surface, then |Aut M | = 4ε(K n) = 2n(n − 1).

Whence,

|Aut M /

α

|= n(n

−1), i.e., Aut M /

α

acts on αX α,β is Frobenius. Applying

Theorem 5.4.10, we know that n is a prime power.

Conversely, if n = pm, let Γ = Zm p , i.e., the additive group in GF (n), where p is

a prime and n a positive integer and let t ∈ Γ generate this multiplicative group. Take

Γ∗ = Γ − 0, where 0 is the identity of Zm p and r : Γ∗ → Γ∗ determined by r ( x) = tx for

x ∈ Γ∗. By definition, we know that r is cyclic permutation on ∆∗. We extend r from Γ∗ to

Γ by defining r (0) = 0. Notice that r ( x + y) = rx + ry for x, y ∈ Γ. Such an extended r is an

automorphism of group Γ. Applying Theorem 5.4.7, we know that Cay M (Γ : Γ∗, r ) ≃ M

is a regular map on orientable surface.

§5.5 CONSTRUCTING REGULAR MAPS BY GROUPS

5.5.1 Regular Tessellation. Let R2 be a Euclidean plane and p, q ≥ 3 be integers. We

know that the angle of a regular p-gon is (1 − 2/ p)π. If q such p-gons fit together around

a common point u ∈ R2, then the angle of p-gons must be 2π/q. Thus

1 −2

p π =2π

q, i.e., ( p

−2)(q

−2) = 4.

We so get three planar regular tessellations of type ( p, q) on a Euclidean plane following:

(4, 4), (3, 6), (6, 3).

For example, a tessellation of type (4, 4) on R2 is shown in Fig.5.5.1.




Fig.5.5.1

Now let S 2 be a sphere. Consider regular p-gons on S 2. The angle of a spherical p-

gon is greater than (1 − 2/ p)π, and gradually increases this value to π if the circum-radius

increases from 0 to π/2. Consequently, if

( p − 2)(q − 2) < 4,

we can adjust the size of the polygon so that the angle is exactly 2π/q, i.e., q such p-gons

will fit together around a common point v ∈ S 2. This fact enables one to get spherical

tessellations of type ( p, q) following:

(2, q), (q, 2), (3, 3), (3, 4), (4, 3), (3, 5), (5, 3).

The type of (2, q) is formed by q lues joining the two antipodal points and the type (q, 2)

is formed by two q-gons, each covering a hemisphere. All of these rest types of spherical

tessellations are the blown up of these five Platonic solids shown in Fig.5.4.1.

Finally, let H 2 be a hyperbolic plane. Consider the regular p-gons on H 2. Then the

angle of such a p-gon is less than (1 − 2/ p)π, and gradually decreases this value to zero if

the circum-radius increases from 0 to ∞. Now if

( p − 2)(q − 2) > 4,

we can adjust the size of the polygon so that the angle is exactly 2 π/q. Thus q such

p-gons will fit together around a common point w ∈ H 2. This enables one to construct

a hyperbolic tessellation of type ( p, q), which is an infinite collection of regular p-gons

filling the hyperbolic plane H 2.

Consider a tessellation of type ( p, q) drawn in thick lines and pick a point in the

interior of each face and call it the ıcenter of the face. In each face, join the center by

dashed and thin line segments with every point covered by q-gons and the midpoint of




every edge, respectively. This structure of tessellation is called the barycentric subdivision

of tessellation. Each of the triangle formed by a thick, a thin and a dashed sides is called

a flag, such as those shown in Fig.5.5.2. Denote all flags of a tessellation by F .

g Xg Zg

Yg

Fig.5.5.2

A tessellation of type ( p, q) is symmetrical by reflection in certain lines, which may

be a successive reflections of three types: X : g → Xg, Y : g → Yg and Z : g → Zg,

where for each flag g, the flag Wg is such the unique flag diff erent from g that shares with

g the thin, the thick or the dashed sides depending on W = X , Y or Z . Obviously,

X 2 = Y 2 = Z 2 = ( XY )2 = (YZ ) p = ( ZX )q = 1 and XY = Y X .

Furthermore, the group X , Y , Z is transitive permutation group on F .

A tessellation of type ( p, q) on surface S is naturally a map M = (X α,β, P) on S

with X α,β = F . The behaviors of X , Y and YZ are more likely to those of β ,α and P on

M . But essentially, X β, Y α and YZ P because X , Y and YZ act on a given g, not

on all g in F . Such X , Y or YZ can be only seen as the localization of β, α or P on a

quadricell g of map M .

5.5.2 Regular Map on Finite Group. Let (Γ; ) be a finite group with presentation

Γ =

x, y, z | x2 = y2 = z2 = ( x y)2 = ( y z) p = ( z x)q = · · · = 1Γ

,

where we assume that all exponents are true orders of the elements and dots indicate a pos-

sible presence of other relations in this subsection. Then a regular map M = M (Γ; x, y, z)

of type ( p, q) on group (Γ; ) is constructed as follows.




Construction 5.5.1 Let g ∈ Γ. Consider a topological triangle, i.e., a flag labeled by g

with its thin, thick and dashed sides labeled by generators x, y and z, respectively. Such

as those shown in Fig.5.5.3.

x

y

zg

Fig.5.5.3

For simplicity, we will identify such flags with their group element labels. Then for each

g ∈ Γ and w ∈ x, y, z, we identify the sides labeled w in the flag g and g w in such a

way that points on the thick, thin or dashed sides meet are identified as well. For example,

such an identification for g = x, y or z is shown in Fig.5.5.4.

g g x x x

y y

z zg g x

g y x

g

g y

z

x

y g z

y

z

g

g z

Fig.5.5.4

This way we get a connected surface S without boundary by Theorem 4.2.2. The cellular

decomposition of S induced by the union of all thick segments forms a regular map M =

M (Γ; x, y, z) of type ( p, q). Such thick segments of S consist of the underlying graph




G( M ) with vertices, edges and faces identified with the left cosets of subgroups generated

by x, y, y, z and z, x in the group (Γ; ), respectively. We therefore get the following

result by this construction.

Theorem 5.5.1 Let (Γ; ) be a finite group with a presentation

Γ =

x, y, z | x2= y2

= z2= ( x y)2

= ( y z) p= ( z x)q

= · · · = 1Γ

.

Then there always exists a regular map M (Γ; x, y, z) of type ( p, q) on (Γ; ).

Consider the actions of left and right multiplication of Γ on flags of M . By Construc-

tion 5.5.1, we have known that the right multiplication by generators x, y and z on a flag

g ∈ Γ gives the permutations X , Y and Z defined in Fig.5.5.2. For the left multiplication

of Γ on flags of M , we have an important result following.

Theorem 5.5.2 Let M = M (Γ; x, y, z) be a regular map of type ( p, q) on a finite group

(Γ; ) , where Γ =

x, y, z | x2 = y2 = z2 = ( x y)2 = ( y z) p = ( z x)q = · · · = 1Γ

. Then

Aut M = LΓ ≃ (Γ; ).

Proof Notice that if two flags F and F ′ are related by a homeomorphism h on S ,

i.e., h : F → F ′, then h : F g → F ′ g. Therefore, the left multiplication preserves

the cell structure of M on S and induces an automorphism of M . Whence, LΓ

≤Aut M .

Now X α,β( M ) = F ( M ) = Γ. By Corollary 5.3.3, there is |Aut M | ≤ |X α,β( M )| = |Γ|.Consequently, there must be Aut M = LΓ. By Theorem 1.2.15, LΓ ≃ (Γ; ). This


There is a simple criterion for distinguishing isomorphic maps M (Γ1; x1, y1, z1) and

M (Γ2; x2, y2, z2) following.

Theorem 5.5.3 Two regular maps M (Γ1; x1, y1, z1) and M (Γ2; x2, y2, z2) are isomorphic if

and only if there is a group isomorphism φ : Γ1 → Γ2 such that φ( x1) = x2, φ( y1) = y2

and φ( z1) = z2.

Proof If there is a group isomorphism φ : Γ1 → Γ2 such that φ( x1) = x2, φ( y1) =

y2 and φ( z1) = z2, we extend this isomorphism φ from flags F ( M (Γ1; x1, y1, z1)) to

F ( M (Γ2; x2, y2, z2)) by

φ(uǫ 11

uǫ 22

· · · uǫ ss ) = φ(uǫ 1

1)φ(uǫ 2

2) · · · φ(uǫ s

s )




for ui ∈ x1, y1, z1, ǫ i ∈ +, − and integers s ≥ 1. Then φ is an isomorphism between

M (Γ1; x1, y1, z1) and M (Γ2; x2, y2, z2) because it preserves the incidence of flags.

Conversely, if φ is an isomorphism from M (Γ1; x1, y1, z1) to M (Γ2; x2, y2, z2), then

it preserves the incidence of vertices, edges and faces. Whence it induces an isomor-phism from flags F ( M (Γ1; x1, y1, z1)) to F ( M (Γ2; x2, y2, z2)), i.e., a group isomorphism

φ : Γ1 → Γ2, which preserve the incidence of vertices, edges and faces if and only if

φ( x1) = x2, φ( y1) = y2 and φ( z1) = z2 by Construction 5.5.1.

Similarly, it can be shown that a regular map M (Γ, x′, y′, z′) is a dual of M (Γ, x, y, z)

if and only if Γ′ = Γ and x′ = y, y′ = x. By this way, regular maps of small genus are

included in the next result.

Theorem 5.5.4 Let M = M (Γ, x, y, z) be a regular map on a finite group Γ.

(A) If M is on the sphere S 2 , then

(1) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)n = ( zx)2 = 1Γ

≃ Dn × Z 2 and M is an

embedded n-dipoles with dual C n on S 2;

(2) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)3 = ( zx)3 = 1Γ

≃ S 4 and M is the tetra-

hedron, which is self-dual on S 2;

(3) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)4 = ( zx)3 = 1Γ

≃ S 4 × Z 2 and M is the

octahedron with dual cube on S 2;

(4) Γ = x, y, z|

x2 = y2 = z2 = ( xy)2 = ( yz)5 = ( zx)2 = 1Γ ≃A5

× Z 2 and M is the

icosahedron with dual dodecahedron on S 2.

(B) If M is on the projective plane P2 , let r = yz and s = zx, then

(1) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)2n = ( zx)3 = zsr n = 1Γ

≃ D2n and M is

the embedded bouquet B2n with dual C 2n on P2;

(2) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)4 = ( zx)3 = zrs−1r 2 s = 1Γ

≃ S 4 and M

is the embedded K (2)

3with dual K 4 on P2 , where K

(2)

3is the graph K 3 with double edges;

(3) Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz)5 = ( zx)3 = zr 2 sr −1 sr −2 s = 1Γ

≃ A5

and M is the embedded K 6 on P2

.

(C) If M is on the torus T 2 , let b, c be integers, then Γ =

r , s | r 4 = s4 = (rs)2 =

(rs−1)b(r −1 s)c = 1Γ

or

r , s | r 6 = s3 = (rs)2 = (rs−1r )b(s−1r 2)c = 1Γ

if bc(b − c) 0

and Γ =

r , s | r 4 = s4 = (rs−1)b(r −1 s)c = 1Γ

or

r , s | r 6 = s3 = (rs−1r )b(s−1r 2)c = 1Γ

if bc(b − c) = 0.

A complete proof of Theorem 5.5.4 can be found in the reference [CoM1]. With the




help of parallel program, orientable regular maps of genus 2 to 15, and non-orientable

regular maps of genus 4 to 30 are determined in [CoD1]. Particularly, the regular maps

on a double-torus or a non-orientable surface of genus 4 are known in the following.

Theorem 5.5.5 M = M (Γ, x, y, z) be a regular map on a finite group Γ , r = yz, s = zx

and t = xr.

(A) If M is orientable of genus 2 , then Γ =

r , s | r 3 = s8 = (rs−3)2 = 1Γ

, or

r , s | r 4 = s6 = (rs−1)2 = 1Γ

, or

r , s | r 4 = s8 = (rs−1)2 = rs3r −1 s−1 = 1Γ

, or

r , s | r 5

= s10 = s2r −3 = 1Γ

, or

r , s | r 6 = s6 = r 2 s−4 = 1Γ

, or

r , s | r 8 = s8 = rs−3 = 1Γ

.

(B) If M is non-orientable of genus 4 , then Γ =

r , s, t | r 4 = s6 = t 2 = ts−1rs−1r −2

= 1Γ

, or r , s, t

|r 4 = s6 = t 2 = (rs−2)2 = s2rs−1r −2t = 1Γ .

We have known that there are regular maps on every orientable surface by Theorem

5.4.2, and there are no regular maps M on non-orientable surfaces of genus 2, 3, 18, 24,

27, 39 and 48 in literature. Whether or not there are infinite non-orientable surfaces

which do not support regular maps is a problem for a long time. However, a general result

appeared in 2004 ([DNS1]), which completely classifies regular maps on non-orientable

surface of genus p + 2 for an odd prime p 3, 7 and 13. For presenting this general result,

let ν( p) be the number of pairs of coprime integers ( j, l) such that j > l > 3, both j and l

are odd and ( j − 1)(l − 1) = p + 1 for a prime p.

Theorem 5.5.6 Let p be an odd prime, p 3, 7, 13 and let N p+2 be a non-orientable

surface of genus p + 2. Then

(1) If p ≡ 1(mod 12) , then there are no regular maps on N p+2;

(2) If p ≡ 5(mod 12) , then, up to isomorphism and duality, there is exactly one

regular map on N p+2;

(3) If p

≡ −5(mod 12) , then, up to isomorphism and duality, there are ν( p) regular

maps on N p+2;

(4) If p ≡ −1(mod 12) , then, up to isomorphism and duality, N p+2 supports exactly

ν( p) + 1 regular maps.

5.5.3 Regular Map on Finite Multigroup. Let P1, P2, · · · , Pn be a family of topological

polygons with even sides for an integer n ≥ 1. Denoted by ∂Pi the boundary of Pi,




1 ≤ i ≤ n. Define a projection π :n

i=1

Pi → (n

i=1

Pi)/ ∼ by

π( x1) π( x2) · · · π( xn) if xi ∈ Pi \ ∂Pi, 1 ≤ i ≤ n,

π( y1) = π( y2) = · · · = π( yn) if yi ∈ ∂Pi, 1 ≤ i ≤ n,

i.e., π is an identification on boundaries of P1, P2, · · · , Pn. Such an identification space

(n

i=1

Pi)/ ∼ is called an m-multipolygon by n polygons and denoted by P. The cross section

of P is shown in Fig.5.5.5(a). Sometimes, a multipolygon maybe homeomorphic to a

surface. For example, the sphere S 2 is in fact a topological multipolygon of 2 polygons

shown in Fig.4.1.2.

It should be noted that the boundary of an m-multipolgon

P is the same as any of

its m-polygon. So we can also get the polygonal presentation of an m-multipolygon such

as we have done in Section 4.2. Similarly, an orientable or non-orientable multisurfaceS is defined on P by identifying side pairs of P. Certainly, S =n

i=1

Pi/ ∼=n

i=1

S i, where

S i = Pi/ ∼ is a surface for integers 1 ≤ i ≤ n. The inclusion mapping πi : S → S i

determined by πi( x) = x for x ∈ S i is called the natural projection of S on S i.

By definition, ∂P/ ∼ is a closed curve on S , called the base line, denoted by LB and

a multisurface S possesses the hierarchical structure, i.e., S \ LB is disconnected union of

Pi \ ∂Pi, 1 ≤ i ≤ n. Such as those shown in Fig.5.5.5(b) for longitudinal and cross section

of a multitorus.

(a)boundary boundary LB LB

(b)

Fig.5.5.5

Similarly considering maps on surface S , we can find such a decomposition of

S

with each components homeomorphic to a open disk of dimensional 2, i.e., a map M onS . So a problem for maps on multisurfaces is presented in the following.

Problem 5.5.1 Determine maps M on S =

ni=1

S i such that πi( M ) is a transitive map,

furthermore a regular map on S i for any integer i, 1 ≤ i ≤ n.

If S is orientable, the answer is affirmed by Theorem 5.4.2 by applying to standard






M =

s j=1

M (Γi j; xi j

, y, zi j)

is such a map that πi j : M → M (Γi j ; xi j , y, zi j ), a regular map on S i j . This fact enables oneto get the following result.

Theorem 5.5.8 For any integers g ≥ 0, n ≥ 1 and p, q ≥ 3 , if there is a regular map

M (Γ; x, y, z) of genus g correspondent to a triangle group Γ =

x, y, z | x2 = y2 = z2 =

( x y)2 = ( y z) p = ( z x)q = 1Γ

, then there is a map M on multisurface S =

ni=1

S i

consisting of n surfaces of genus g such that πi( M ) is a regular map M (Γi; xi, y, zi) , par-

ticularly, there is a map

M on

S such that πi(

M ) = M (Γ; x, y, z) for integers 1 ≤ i ≤ n.

§5.6 REMARKS

5.6.1 A topological map M is essentially a decomposition of a surface S with com-

ponents homeomorphic to 2-disk, which can be also characterized by the embedding of

graph G[ M ] on S . Many mathematicians had contributed to the foundation of map theory,

such as those of Tutte in [Tut1], Jones and Singerman in [JoS1], Vince in [Vin1]-[Vn2]

and Bryant and Singerman in [BrS1] characterizing a map by qurdricells or flags. Theyare essentially equivalent. There are many excellent books on these topics today. For

example, [GrT1] and [Whi1] on embedding and topological maps, [MoT1] on the topo-

logical behavior of embeddings and [Liu2]-[Liu4] on algebraic maps with enumerative

theory.

5.6.2 Although it is difficult to determine the automorphism group of a graph in general,

it is easy to find the automorphism group of a map. By Theorem 5 .3.6, the automorphism

group of map M = (X α,β, P) is the centralizer of the group α, β, P in the symmetric

group S X α,β . In fact, there is an efficient algorithm for getting an automorphism group

of map with complexity not bigger than O(ε2( M )). See [Liu1], [Liu3]-[liu4] for details.

Besides, a few mathematicians also characterized automorphism group of map by that of

its underlying graph. This enables one to know that the automorphism group of map is an

extended action subgroup of the semi-arc automorphism group of its underlying graph.

See also [Mao2] and [MLW1] for details.



Sec.5.6 Remarks 209

5.6.3 The research of regular maps, beginning for searching stellated polyhedra of sym-

metrical beauty, is more early than that of general map, which appeared firstly in the work

of Kepler in 1619. The well-known such polyhedra are the five Platonic polyhedra. There

are two equivalent definitions for regular map by let the automorphism group of map M transitive on its quadricells or flags. Both of them makes the largest possible on auto-

morphisms of a map, i.e., transitive and fixed-free. This enables one knowing that the

automorphism group of a map is transitive on its vertices, edges and faces, and also its

upper bound of regular maps of genus≥ 2. For many years, one construct regular maps

by that of symmetric graphs, such as those of Cayley graphs, complete graphs, cubic

graph and Paley graph on surfaces. The materials in references [Big1]-[Big2], [BiW1]

and [JaJ1] are typical such examples.

Such as those discussions in the well-know book [CoM1] on discrete group withgeometry. A more efficient way for constructing regular map is by that of the triangle

group Γ =

x, y, z | x2 = y2 = z2 = ( xy)2 = ( yz) p = ( zx)q = 1Γ

. In fact, by the barycentric

subdivision of map on surface, a regular map M is unique correspondent to a triangle

group Γ and vice vera. This correspondence turns the question of finding regular maps to

that of classifying or constructing such triangle groups and enables one to classify regular

maps of small genus. For example, the classification of regular maps on N p+2 for an odd

prime p in [DNS1] is by this way, and the classification of regular maps for orientable

genus from 2 to 15, non-orientable from 4 to 30 in [CoD1] is also by this way with thehelp of parallel program.

5.6.4 A multisurface S is introduced for characterizing hierarchical structures of topo-

logical space. Besides this structure, its base line LB is common and the same as that of

standard surface O p or N q. We have shown that there is a map M on S such that its projec-

tion on any surface of S is a regular map by applying Cayley maps on finite groups, and

by regular maps on finite triangle group. Besides for regular map, we can also consider

embedding question on multisurface

S . Since all genus of surface in a multisurface

S is

the same, we define the genus g(S ) of S to be the genus of its surface.Let G be a connected graph. Define its orientable or non-orientable genusγ Om(G),γ N

m(G) on multisurface S consisting of m surfaces S by

γ Om(G) = min g(S ) | G is 2 − cell embeddable on orinetable multisurface S ,

γ N m(G) = min g(S ) | G is 2 − cell embeddable on orinetable multisurface S .




Then we are easily knowing that γ O1

(G) = γ (G) and γ N 1

(G) = γ (G) by definition. The

problems for embedded graphs following are particularly interesting for researchers.

Problem 5.6.1 Let n, m

≥1 be integers. Determine γ Om(G) and γ N

m(G) for a connected

graph G, particularly, the complete graph K n and the complete bipartite graph K n,m.

Problem 5.6.2 Let G be a connected graph. Characterize the embedding behavior of G

on multisurface S , particularly, those embeddings whose every facial walk is a circuit,

i.e, a strong embedding of G on S .

The enumeration of non-isomorphic objects is an important problem in combina-

torics, particular for maps on surface. See [Liu2] and [Liu4] for details. Similar problems

for multisurface are as follows.

Problem 5.6.3 Let S be a multisurface. Enumerate embeddings or maps on S by param-

eters, such as those of order, size, valency of rooted vertex or rooted face, · · ·.Problem 5.6.4 Enumerate embeddings on multisurfaces for a connected graph G.

For a connected graph G, its orientable, non-orientable genus polynomial gm[G]( x),gm[G]( x) is defined to be

gm[G]( x) =i≥0

gOmi(G) xi and

gm[G]( x) =

i≥0

g N mi(G) xi,

where gOmi

(G), g N mi

(G) are the numbers of G on orientable or non-orientable multisurfaceS consisting of m surfaces of genus i.

Problem 5.6.5 Let m ≥ 1 be an integer. Determine gm[G]( x) and gm[G]( x) for a connected

graph G, particularly, for the complete or complete bipartite graph, the cube, the ladder,

the bouquet, · · ·.



CHAPTER 6.

Lifting Map Groups

The voltage assignment technique on graphs or maps is in fact a constructionof regular coverings of graphs or maps, i.e., covering spaces in lower dimen-

sional cases. For such covering spaces, an interesting problems is that finding

conditions on the assignment so that an automorphism of graph or map is also

an automorphism of the lifted graph or map, and then apply this technique

to finding regular maps or solving problems on Klein surfaces. For these ob-

jectives, we introduce topological covering spaces, covering mappings first,

and then voltage graphs and maps in Section 6.1. The lifting map group is

discussed in the following section. These conditions such as those of locally

invariant, A J -uniform and A J -compatible, and furthermore, a condition for a

finite group to be that of a map by voltage assignment can be found in Section

6.2, which enables one finding a formulae related the Euler-Poincare charac-

teristic with parameters on maps or its quotient maps. These formulae enables

us to discussing the minimum or maximum order of automorphisms of a map,

i.e., conformal transformations realizable by maps M on Riemann or Klein

surfaces in Section 6.5. Section 6.4 presents a combinatorial generalization of

the famous Hurwitz theorem on orientation-preserving automorphism groups

of Riemann surfaces, which enables us to get the upper or lower bounds of

automorphism groups of Klein surfaces. All these discussions support a con-

jecture in forewords of Chapter 5 in [Mao2], i.e., CC conjecture discussed in

the last chapter of this book.



212 Chap.6 Lifting Map Groups

§6.1 VOLTAGE MAPS

6.1.1 Covering Space. Let S be a topological space. A covering space

S of S consisting

of a space S with a continuous mapping p : S →S such that any point x

∈S possesses an

arcwise connected neighborhood U x, and any arcwise connected component of p−1(U x)

is mapped topologically onto U x by p. Such an opened neighborhoods U x is called an

elementary neighborhood and p a projection from S to S .

Definition 6.1.1 Let S , T be topological spaces, x0 ∈ S , y0 ∈ T and f : (T , y0) → (S , x0)

a continuous mapping. If (S , p) is a covering space of S , x0 ∈ S , x0 = p( x0) and there

exists a mapping f l : (T , y0) → (S ,

x0) such that f = f l p, then f l is a lifting of f ,

particularly, if f is an arc, f l is called a lifting arc.

The following result asserts the lifting of an arc is uniquely dependent on the initial

point.

Theorem 6.1.1 Let (S , p) be a covering space of S , x0 ∈ X and p( x0) = x0. Then there

exists a unique lifting arc f l : I → S with initial point x0 for each arc f : I → S with

initial point x0.

A complete proof of Theorem 6.1.1 can be found in references [Mas1] or [Mun1],

which applied the property of Lebesgue number on metric space.

Theorem 6.1.2 Let (S , p) be a covering space of S , x0 ∈ S and p( x0) = x0. Then

(1) the induced homomorphism p∗ : π(S , x0) → π(S , x0) is a monomorphism;

(2) for x ∈ p−1( x0) , the subgroups p∗π(S , x0) are exactly a conjugacy class of sub-

groups of π(S , x0).

Proof Applying Theorem 6.1.1, for x0 ∈ S and p( x0) = x0, there is a unique mapping

on loops from S with base point

x0 to S with base point x0. Now let Li : I → S , i = 1, 2

be two arcs with the same initial point

x0 in

S . We prove that if pL1 ≃ pL2, then L1 ≃ L2.

Notice that pL1 ≃ pL2 implies the existence of a continuous mapping H : I × I → S

such that H (s, 0) = pl1(s) and H (s, 1) = pL2(s). Similar to the proof of Theorem 3.10, we

can find numbers 0 = s0 < s1 < · · · < sm = 1 and 0 = t 0 < t 1 < · · · < t n = 1 such that each

rectangle [si−1, si] × [t j−1, t j] is mapped into an elementary neighborhood in S by H .

Now we construct a mapping G : I × I → S with pG = H , G(0, 0) = x0 hold by the

following procedure.



Sec.6.1 Voltage Maps 213

First, we can choose G to be a lifting of H over [0, s1] × [0, t 1] since H maps this

rectangle into an elementary neighborhood of p(

x0). Then we extend the definition of G

successively over the rectangles [si−1, si] × [0, t 1] for i = 2, 3, · · · , m by taking care that it

is agree on the common edge of two successive rectangles, which enables us to get G overthe strip I × [0, t 1]. Similarly, we can extend it over these rectangles I × [t 1, t 2], [t 2, t 3], · · ·,etc.. Consequently, we get a lifting H l of H , i.e., L1 ≃ L2 by this construction.

Particularly, if L1 and L2 were two loops, we get the induced monomorphism homo-

morphism p∗ : π(S , x0) → π(S , x0). This is the assertion of (1).

For (2), suppose x1 and x2 are two points of S such that p( x1) = p( x2) = x0. Choose

a class L of arcs in S from

x1 to

x2. Similar to the proof of Theorem 3.1.7, we know that

L = L[a] L−1, [a] ∈ π(

S ,

x1) defines an isomorphism L : π(

S ,

x1) → π(

S ,

x2). Whence,

p∗(π(S , x1)) = p∗( L)π(S , x2) p∗( L−1). Notice that p∗( L) is a loop with a base point x0. Weknow that p∗( L) ∈ π(S , x0), i.e., p∗π(S , x0) are exactly a conjugacy class of subgroups of

π(S , x0).

Theorem 6.1.3 If (S , p) is a covering space of S , then the sets p−1( x) have the same

cardinal number for all x ∈ S .

Proof For any points x1 and x2 ∈ S , choosing an arc f in S with initial point x1 and

terminal point x2. Applying f , we can define a mapping Ψ : p−1( x1) → p−1( x2) by the

following procedure.For ∀ y1 ∈ p−1( x1), we lift f to an arc f l in S with initial point y1 such that p f l = f .

Denoted by y2 the terminal point of f l. Define Ψ( y1) = y2.

By applying the inverse arc f −1, we can define Ψ−1( y2) = y1 in an analogous way.

Therefore, ψ is a 1 − 1 mapping form p−1( x1) to p−1( x2).

Usually, this cardinal number of the sets p−1( x) for x ∈ S is called the number of

sheets of the covering space (S , p) on S . If | p−1( x)| = n for x ∈ S , we also say it an

n-sheeted covering.

6.1.2 Covering Mapping. Let M = ( X α,β, P) and M = (X α,β, P) be two maps. The

map M is called to be covered by map M if there is a mapping π : X α,β → Xα,β such that

∀ x ∈ Xα,β,

απ( x) = πα( x), βπ( x) = πβ( x) and π P( x) = Pπ( x).

Such a mapping π is called a covering mapping. For ∀ x ∈ X α,β, define the quadricell set






and ∀ x1, x2 ∈ X α,β, if x1 x2, then πgπ−1( x1) πgπ−1( x2). Otherwise, let

πgπ−1( x1) = πgπ−1( x2) = x0 ∈ X α,β.

Then we must have that x1 = πg−1π−1( x0) = x2, which contradicts to the assumption.

By definition, for x ∈ X α,β we have that

π∗α( x) = πgπ−1α( x) = πgαπ−1( x) = παgπ−1( x) = απgπ−1( x) = απ∗( x),

π∗ β( x) = πgπ−1 β( x) = πg βπ−1( x) = πβgπ−1( x) = βπgπ−1( x) = βπ∗( x).

Now π( P) = P . We therefore get that

π∗P( x) = πgπ−1P( x) = πg Pπ−1( x) = π Pgπ−1( x) = Pπgπ−1( x) =

Pπ∗( x).

Consequently, πgπ−1 ∈ Aut M , i.e., π∗ : Aut M → Aut M .

Now we prove that π∗ is a homomorphism from Aut M to Aut M . In fact, for ∀g1, g2 ∈Aut M , we have that

π∗(g1g2) = π(g1g2)π−1= (πg1π−1)(πg2π−1) = π∗(g1)π∗(g2).

Whence, π∗ : Aut

M → Aut M is a homomorphism.

6.1.3 Voltage Map with Lifting. Let G be a connected graph and (Γ;

) a group. For

each edge e ∈ E (G), e = uv, an orientation on e is such an orientation on e from u to

v, denoted by e = (u, v), called the plus orientation and its minus orientation, from v

to u, denoted by e−1 = (v, u). For a given graph G with plus and minus orientation on

edges, a voltage assignment on G is a mapping σ from the plus-edges of G into a group Γ

satisfying σ(e−1) = σ−1(e), e ∈ E (G). These elements σ(e), e ∈ E (G) are called voltages,

and (G, σ) a voltage graph over the group (Γ; ).

For a voltage graph (G, σ), its lifting Gσ = (V (Gσ), E (Gσ); I (Gσ)) is defined by

V (Gσ) = V (G) × Γ, (u, a) ∈ V (G) × Γ abbreviated to ua;

E (Gσ) = (ua, vab)|e+= (u, v) ∈ E (G), σ(e+) = b

and

I (Gσ) = (ua, vab)| I (e) = (ua, vab) i f e = (ua, vab) ∈ E (Gσ).




This is a |Γ|-sheet covering of the graph G. For example, let G = K 3 and Γ = Z 2.

Then the voltage graph (K 3, σ) with σ : K 3 → Z 2 and its lifting are shown in Fig.6.1.1.

u

w

10

0

(G, σ)

v

u0

u1

v0

v1

w0

w1

Gσ

Fig.6.1.1

We can find easily that there is a unique lifting path in Γl with an initial point

x for

each path with an initial point x in Γ, and for ∀ x ∈ Γ, | p−1( x)| = 2.For finding a homomorphism between Klein surfaces, voltage maps are extensively

used, which is introduced by Gustin in 1963 and extensively used by Youngs in 1960s for

proving the Heawood map coloring theorem and generalized by Gross in 1974 ([GrT1]).

By applying voltage graphs, the 2-factorable graphs are enumerated in [MaT2] also.

Now we present a formally algebraic definition for voltage maps, not using geomet-

rical intuition following.

Definition 6.1.2 Let M = (X α,β, P) be a map and (Γ;

) a finite group. A pair ( M , ϑ) is a

voltage map with group (Γ; ) if ϑ : X α,β → Γ , satisfying conditions following:

(1) For ∀ x ∈ X α,β, ϑ(α x) = ϑ( x), ϑ(αβ x) = ϑ( β x) = ϑ−1( x);

(2) For ∀F = ( x, y, · · · , z)( β z, · · · , β y, β x) ∈ F ( M ) , the face set of M, ϑ(F ) =

ϑ( x)ϑ( y) · · · ϑ( z) and ϑ(F )|F ∈ F (u), u ∈ V ( M ) = Γ , where F (u) denotes all the faces

incident with vertex u.

For a voltage map ( M , ϑ), define

X αϑ,βϑ = X α,β

×Γ,

P ϑ =

( x, y,···, z)(α z,···,α y,α x)∈V ( M )

g∈Γ

( xg, yg, · · · , zg)(α zg, · · · , α yg, α xg)

and

αϑ =

x∈X α,β, g∈Γ

( xg, α xg), βϑ =

x∈X α,β , g∈Γ

( xg, β xgϑ( x)),

where ug denotes the element (u, g) ∈ X α,β × Γ.




Then it can be shown immediately that M ϑ = (Xαϑ,βϑ , P ϑ) also satisfies the condi-

tions of map, and with the same orientation as map M . Whence, we define the lifting map

of a voltage map in the following definition.

Definition 6.1.3 Let ( M , ϑ) be a voltage map with group (Γ; ). Then the map M ϑ =

(Xϑα,β

, P ϑ) is defined to be the lifting map of ( M , ϑ).

There is a natural projection π : M ϑ → M from the lifted map M ϑ to M by π( xg) = x

for ∀g ∈ Γ and x ∈ X α,β( M ), which means that M ϑ is a |Γ|-cover M . Denote by

π−1( x) = xg ∈ X α,β( M ϑ) | g ∈ Γ ,

called the fiber over x ∈ X α,β( M ). For a vertex v = (C )(αC α−1) ∈ V ( M ), let C denote

the set of quadricells in cycle C . Then the following result is obvious by definition.

Theorem 6.1.7 The numbers of vertices and edges in a lifting map M ϑ of voltage map

( M , ϑ) with group (Γ; ) are respectively

ν( M ϑ) = ν( M )|Γ| and ε( M ϑ) = ε( M )|Γ|.

Theorem 6.1.8 Let F = (C ∗)(αC ∗α−1) be a face in the map M. Then there are |Γ|/o(F )

faces in the lifting map M ϑ with group (Γ; ) of length |F |o(F ) lifted from the face F,

where o(F ) denotes the order of x∈C ϑ( x) in group (Γ;

).

Proof Let F = (u, v · · · , w)( βw, · · · , βv, βu) be a face in the map M and k is the length

of F . Then, for ∀g ∈ Γ the conjugate cycles

(C ∗)ϑ = (ug, vgϑ(u), · · · , ugϑ(F ), vgϑ(F )ϑ(u), · · · , wgϑ(F )2 , · · · , wgϑo(F )−1(F ))

β(ug, vgϑ(u), · · · , ugϑ(F ), vgϑ(F )ϑ(u), · · · , wgϑ(F )2 , · · · , wgϑo(F )−1(F ))−1 β−1.

is a face in M ϑ with length ko(F ) by definition. Therefore, there are |Γ|/o(F ) faces in the

lifting map M ϑ.

We therefore get the Euler-Poincare characteristic of a lifted map following.

Theorem 6.1.9 The Euler-Poincar´ e characteristic χ( M ϑ) of the lifting map M ϑ of a volt-

age map ( M , ϑ) with group (Γ; ) is

χ( M ϑ) = |Γ|( χ( M ) +

m∈O(F ( M ))

(−1 +1

m)),




where O(F ( M )) denotes the set of faces in M of order o(F ).

Proof According to the Theorems 6.1.7 and 6.1.8, the lifting map M ϑ has |Γ|ν( M )

vertices,

|Γ

|ε( M ) edges and

|G

| m∈O(F ( M ))

1

m

faces. Therefore, we know that

χ( M ϑ) = ν( M ϑ) − ε( M ϑ) + φ( M ϑ)

= |Γ|ν( M ) − |Γ|ε( M ) + |Γ|

m∈O(F ( M ))

1

m

= |G|( χ( M ) − φ( M ) +

m∈O(F ( M ))

1

m)

= |G|( χ( M ) +

m∈O(F ( M ))

(−1 +1

m)).

§6.2 GROUP BEING THAT OF A MAP

6.2.1 Lifting Map Automorphism. Let ( M , σ) be a voltage map with σ : X α,β → Γ, u ∈V ( M ) and W = x1 x2 · · · xk a walk encoded by the corresponding sequence of quadricells

xi, i = 1, 2, · · · , k in M , i.e., the qudricell after xi is Pαβ xi by the traveling ruler on M .

Define the net voltage on W to be the product

σ(W ) = σ( x1) σ( x2) · · · σ( xk )

and the local voltage group Γ(u) by

Γ(u) = σ(W ) | W is a closed walk based at a quadricell u .

By Definition 6.1.2, we know that Γ(u) = Γ for ∀u ∈ X α,β( M ). For x ∈ X α,β, denote

by Π( M , x) the set of all such closed walks based at x. Then Π( M , x) = π1( M , x), the

fundamental group of M based at x.

Let σ1, σ2 : X α,β → Γ be two voltage assignments on a map M = (X α,β, P) and

id M an identity transformation on X α,β, i.e., both of M σ1 and M σ2 are |Γ|-covers of M with

natural projections π1 : M σ1 → M and π2 : M σ2 → M on M . Then we know

X α,β( M σ1 ) = X α,β( M σ2 ) = xg | x ∈ X α,β( M ), g ∈ Γ

by definition. Then σ1, σ2 are said to be equivalent if there exists an isomorphism τ :

M σ1 → M σ2 that makes the following diagram



Sec.6.2 Group being That of a Map 219

¹

M σ1 M σ2

M

τ

M ¹

id M

π1 π2

commutate. The following result is fundamental.

Theorem 6.2.1 Let σ1, σ2 : X α,β → Γ be two voltage assignments on a map M =

(X α,β, P) , u ∈ X α,β( M ). Then σ1, σ2 are equivalent if and only if there exists an auto-

morphism τ of group Γ such that

τσ1(W ) = σ2(W )

for every closed walk W in M based at u.

Proof Choose a closed walk W in map M based at u. If σ1 and σ2 are equivalent,

then there exists an automorphism τ : M σ1 → M σ2 such that τ(W σ1 ) = W σ2 . Define

τ∗ : Γ → Γ by τ∗ : τσ1(W ) → σ2(W ). Let W ′ be another closed walk in M based at u.

Notice that WW ′ is also a closed walk based at u in M . We find that

τσ1(WW ′) = τσ1(W )τσ1(W ′) = σ2(W )σ2(W ′),

i.e., τ∗(σ1(W )σ1(W ′)) = τ∗(σ1(W ))τ∗(σ1(W ′)). Thus τ∗ is an automorphism of Γ. By

definition, we are easily get that τ∗σ1(W ) = σ2(W ).

Conversely, if there exists an automorphism τ′ ∈ AutΓ such that τ′σ1(W ) = σ2(W )

for every closed walk W in M based at u, let τ : X α,β( M σ1 ) → X α,β( M σ1 ) be determined

by τ : W τ′σ1 → W σ2 , i.e, τ′σ1W (τ′σ1)−1 = σ2W σ−1

2 . Then it is easily to know that

τ (Pαβ)σ1 τ−1 = (τ′σ1) ( x,···, z)(α z,···,α x)∈V ( M ), g∈Γ

( xg, · · · , zg)(α zg, · · · , α xg) (τ′σ1)−1

=

( x, y,···, z)(α z,···,α y,α x)∈V ( M ), g∈Γ

τ′σ1( xg, yg, · · · , zg)(α zg, · · · , α yg, α xg)(τ′σ1)−1

=

( x,···, z)(α z,···,α x)∈V ( M ), g∈Γ

σ2( xg, yg, · · · , zg)(α zg, · · · , α yg, α xg)σ−12

= (Pαβ)σ2




i.e.,

P σ1 τ = τPσ2

and

ασ1 τ = τασ2 , βσ1 τ = τβσ1 .

Thus τ is an isomorphism from M σ1 to M σ2 by definition. Whence, we know that σ1 and

σ2 are equivalent.

Such an isomorphism τ from M σ1 to M σ2 induced by an automorphism τ′ of M is

called a lifted isomorphism of τ′. Particularly, if σ1 = σ2 = σ, a lifted isomorphism from

M σ1 to M σ2 is called a lifted automorphism of τ′. Theorem 6.2.1 enables one to get the

following result.

Theorem 6.2.2 An automorphism φ of voltage map M with assignment σ → Γ is a lifted

automorphism of map M σ if and only if every closed walk W with net voltage σ(W ) = 1Γ

implies that σ(φ(W )) = 1Γ in ( M , σ).

Furthermore, let M = (X α,β, P) be a map, (Γ; ) a finite group and A ≤ Aut M , a

map group. We say that a voltage assignment σ : X α,β → Γ is locally A -invariant at a

quadricell u if, for ∀τ ∈ A and every walk W ∈ Π( M , u), we have

σ(W ) = 1Γ ⇒ σ(τ(W )) = 1Γ.

Particularly, a voltage assignment is locally τ-invariant for τ ∈ Aut M if it is locally in-

variant respect to the group τ generated by τ. Then Theorem 6.2.2 implies the following

conclusion.

Corollary 6.2.1 Let M = (X α,β, P) be a map with a voltage assignment σ : X α,β → Γ ,

π : M σ → M and A ≤ Aut M. Then A ≤ Aut M σ if and only if σ is locally A -invariant.

Notice that a map M = (X α,β, P) is regular if |Aut M | = |X α,β|. We know the

following result by Corollary 6.2.1.

Corollary 6.2.2 Let M be a regular map with a locally Aut M-invariant voltage assign-

ment σ : X α,β → Γ. Then M σ is also regular.

Proof Notice that the action g : uh → ugh naturally induced an automorphism on

fiber π−1(u) of M σ for ∀u ∈ α,β and g ∈ Γ. Now all automorphisms of M are lifted to M σ.

Whence, |Aut M σ| = |Γ||Aut M | = 4|Γ|ε( M ) = |X α,β( M σ)|. Thus M σ is a regular map.




6.2.2 Map Exponent Group. Let M = (X α,β, P) be a map. An integer k is an exponent

of M if the map M k = (X α,β, Pk ) is isomorphic to M , i.e., there exists a permutation τ on

X α,β such that τα = ατ, τβ = βτ and τP k = Pτ. Such a permutation τ ∈ Aut 12

G[ M ] is

called an isomorphism associated with exponent k .If k is an exponent of M , then Pk is also a basic permutation on X α,β with Axioms

1 − 2 hold. So gcd(k , ρ M (v)) = 1 for v ∈ V ( M ). Consequently, k must be a coprime with

the order o(P) of P, the least common multiple of valencies of vertices in M .

Obviously, 1 is an exponent of M . On the other hand, the integer −1 is an exponent

if M is isomorphic to its mirror (X α,β, P−1). Now let l ≡ k (modo(P)) and k an exponent

of M . Then P l = Pk . Thus l is also an exponent of M . Let k , l be two exponents

associated with isomorphisms τ, θ , respectively. Then

Pklθτ = (Pk )lθτ = θ P lτ = θτP,

i.e., kl is also an exponent of M associated with isomorphism θτ ∈ Aut 12

G[ M ]. We

therefore find the following result.

Theorem 6.2.3 Let M be a map. Then all residue classes of exponents mod(o(P)) of M

form a group, and all isomorphisms associated with exponents of M form a subgroup of

Aut 12

G[ M ] , denoted by Ex( M ) and Exo( M ) , respectively.

Now let (Γ;

) be a finite group and let ι : Γ

→Ex( M ), Ψ : Exo( M )

→Ex( M ) be

homomorphisms with KerΨ = Aut M = A. Denote by A J = Ψ−1( J ), where J = ι(Γ). Then

the derived map M σ,ι is a map (X ασ,ι,βσ,ι, P σ,ι) with

X ασ,ι,βσ,ι = X α,β × Γ

and

Pσ,ι =

( x, y,···, z)(α z,···,α y,α x)∈V ( M ), g∈Γ

( xg, yg, · · · , zg)(α zg, · · · , α yg, α xg)

ι(g),

ασ,ι = x∈X α,β, g∈Γ

( xg, α xg), βσ,ι = x∈X α,β , g∈Γ

( xg, β xgϑ( x)).

A voltage assignment σ : X α,β( M ) → Γ is called A J -uniform if for every u-based

closed walk W on M with σ(W ) = 1Γ and every isomorphism τ ∈ A J , one has σ(τ(W )) =

1Γ. Similarly, an exponent homomorphism τ of M is A J -compatible with σ if for every

u-based walk W and every τ ∈ A J , one always has ισ(W ) = ισ(τ(W )). Then we have the

following result.




Theorem 6.2.4 Let M be an orientable regular map, σ : X α,β( M ) → Γ a voltage assign-

ment and ι : Γ → Ex( M ) with ι(Γ) = J. Then M σ,ι is an orientable regular map if σ is

A J -uniform and τ is A J -compatible with σ.

A complete proof of theorem 6.2.4 was established in [NeS2]. Certainly, the reader

can find more results on constructing regular maps by graphs in [NeS1]-[NeS2].

6.2.3 Group being That of a Lifted Map. A permutation group Γ action on Ω is called

fixed-free if Γ x = 1Γ for ∀ x ∈ Ω. We have the following result on fixed-free permutation

group.

Lemma 6.2.1 Any automorphism group Γ of a map M = (X α,β, P) is fixed-free on X α,β.

Proof Notice that Γ

≤Aut M , we get that Γ x

≤(Aut M ) x for

∀ x

∈X α,β. We have

known that (Aut M ) x = 1Γ. Whence, there must be that Γ x = 1Γ, i.e., Γ is fixed-free.

Notice that the automorphism group of a lifted map has a obvious subgroup deter-

mined by the following lemma.

Lemma 6.2.2 Let M ϑ be a lifted map of a voltage assignment ϑ : X α,β → Γ. Then Γ is

isomorphic to a fixed-free subgroup of Aut M ϑ on V ( M ϑ).

Proof For ∀g ∈ Γ, we prove that the induced action g∗ : X αϑ,βϑ → X αϑ,βϑ by

g∗ : xh → xgh is an automorphism of map M ϑ.

In fact, g∗ is a mapping on X αϑ,βϑ and for ∀ xu ∈ X αϑ,βϑ , we know that g∗ : xg−1u → xu.

Now if for xh, y f ∈ X αϑ,βϑ , xh y f , we have that g∗( xh) = g∗( y f ). Thus xgh = yg f

by the definition. So we must have x = y and gh = g f , i.e., h = f . Whence, xh = y f ,

contradicts to the assumption. Therefore, g∗ is 1 − 1 on X αϑ,βϑ .

We prove that for xu ∈ X αϑ,βϑ, g∗ is commutative with αϑ, βϑ and Pϑ. Notice that

g∗αϑ xu = g∗(α x)u = (α x)gu = α xgu = αg∗( xu);

g∗ βϑ( xu) = g∗( β x)uϑ( x) = ( β x)guϑ( x) = β xguϑ( x) = βϑ( xgu) = βϑg∗( xu)

and

g∗P ϑ( xu)

= g∗

( x, y,···, z)(α z,···,α y,α x)∈V ( M )

u∈G

( xu, yu, · · · , zu)(α zu, · · · , α yu, α xu)( xu)

= g∗ yu = ygu




=

( x, y,···, z)(α z,···,α y,α x)∈V ( M )

gu∈G

( xgu, ygu, · · · , zgu)(α zgu, · · · , α ygu, α xgu)( xgu)

= Pϑ( xgu) = Pϑg∗( xu).

Therefore, g∗ is an automorphism of the lifted map M ϑ.

To see that g∗ is fixed-free on V ( M ), choose ∀u = ( xh, yh, · · · , zh)(α zh, · · · , α yh, α xh) ∈V ( M ), h ∈ Γ. If g∗(u) = u, i.e.,

( xgh, ygh, · · · , zgh)(α zgh, · · · , α ygh, α xgh) = ( xh, yh, · · · , zh)(α zh, · · · , α yh, α xh),

assume that xgh = wh, where wh ∈ xh, yh, · · · , zh, α xh, α yh, · · · , α zh. By definition, there

must be that x = w and gh = h. Therefore, g = 1Γ, i.e., ∀g ∈ Γ, g∗ is fixed-free on V ( M ).

Define τ : g∗

→g. Then τ is an isomorphism between the action of elements in Γ on

X αϑ,βϑ and the group Γ itself.

According to Lemma 6.2.1, for a given map M and a group Γ ≤ Aut M , we define a

quotient map M /Γ = (X α,β/Γ, P/Γ) as follows.

X α,β/Γ = xΓ| x ∈ X α,β,

where xΓ denotes the orbit of Γ action on x in X α,β and

P/Γ = ( x, y,···, z)(α z,···,α y,α x)∈V ( M )

( xΓ, yΓ,

· · ·)(

· · ·, α yΓ, α xΓ)

since Γ action on X α,β is fixed-free.

Such a map M may be not a regular covering of its quotient M /Γ. We have the

following result characterizing fixed-free automorphism groups of map on V ( M ).

Theorem 6.2.5 An finite group (Γ; ) is a fixed-free automorphism group of map M =

(X α,β, P) on V ( M ) if and only if there is a map ( M /Γ, Γ) with a voltage assignment

ϑ : X α,β/Γ → Γ such that M ( M /Γ)ϑ.

Proof The necessity of the condition is already proved in the Lemma 2.2.2. We only

need to prove its sufficiency.

Denote by π : M → M /Γ the quotient mapping from M to M /Γ. For each element of

π−1( xΓ), we give it a label. Choose x ∈ π−1( xΓ). Assign its label l : x → x1Γ, i.e., l( x) = x1Γ

.

Since the group Γ acting on X α,β is fixed-free, if u ∈ π−1( xΓ) and u = g( x), g ∈ Γ, we label

u with l(u) = xg. Whence, each element in π−1( xΓ) is labeled by a unique element in Γ.




Now we assign voltages on the quotient map M /Γ = (X α,β/Γ, P/Γ). If β x = y, y ∈π−1( yΓ) and the label of y is l( y) = y∗

h, h ∈ Γ, where, l( y∗) = 1Γ, then we assign a voltage

h on xΓ,i.e., ϑ( xΓ) = h. We should prove this kind of voltage assignment is well-done,

which means that we must prove that for ∀v ∈ π−1

( xΓ

) with l(v) = j, j ∈ Γ, the label of βvis l( βv) = jh. In fact, by the previous labeling technique, we know that the label of βv is

l( βv) = l( βgx) = l(g β x) = l(gy) = l(ghy∗) = gh.

Denote by M l the labeled map M on each element in X α,β. Whence, M l M . By the

previous voltage assignment, we also know that M l is a lifting of the quotient map M /Γ

with the voltage assignment ϑ : X α,β/Γ → Γ. Therefore,

M ( M /Γ)ϑ.


According to the Theorem 6.2.5, we get the following result for a group to be a map

group.

Theorem 6.2.6 If a group Γ ≤ Aut M is fixed-free on V ( M ) , then

|Γ|( χ( M /Γ) +

m∈O (F ( M /Γ))

(−1 +1

m)) = χ( M ).

Proof By the Theorem 6.2.5, we know that there is a voltage assignment ϑ on the

quotient map M /Γ such that

M ( M /Γ)ϑ.

Applying Theorem 6.1.9, we know the Euler characteristic of map M is

χ( M ) = |Γ|( χ( M /Γ) +

m∈O(F ( M /Γ))

(−1 +1

m)).

Theorem 6.2.6 has some applications for determining the automorphism group of a

map such as those of results following.

Corollary 6.2.3 If M is an orientable map of genus p, Γ ≤ Aut M is fixed-free on V ( M )

and the genus of the quotient map M /Γ is γ , then

|Γ| =2 p − 2

2γ − 2 +

m∈O (F ( M /Γ))

(1 − 1m

)).




Particularly, if M /Γ is planar, then

|Γ| =2 p − 2

−2 + m∈

O (F ( M /Γ))

(1 − 1m

)).

Corollary 6.2.4 If M is a non-orientable map of genus q, Γ ≤ Aut M is fixed-free on

V ( M ) and the genus of the quotient map M /Γ is δ , then

|Γ| =q − 2

δ − 2 +

m∈O (F ( M /Γ))

(1 − 1m

)).

Particularly, if M /Γ is projective planar, then

|Γ| =

q

−2

−1 + m∈O (F ( M /Γ))

(1 − 1m

)) .

By applying Theorem 6.2.5, we can also find the Euler characteristic of the quotient

map, which enables us to get the following result for a group being that of map.

Theorem 6.2.7 If a group Γ ≤ Aut M, then

χ( M ) +

g∈Γ,g1Γ

(|Φv(g)| + |Φ f (g)|) = |Γ| χ( M /Γ),

where, Φv(g) = v|v ∈ V ( M ), vg = v, Φ f (g) = f | f ∈ F ( M ), f g = f , and if Γ is fixed-free

on V ( M ), then

χ( M ) +

g∈Γ,g1Γ

|Φ f (g)| = |Γ| χ( M /Γ).

Proof By the definition of quotient map, we know that

φv( M /Γ) = orbv(Γ) =1

|Γ|g∈Γ

|Φv(g)|

and

φ f ( M /Γ) = orb f (Γ) =1

|Γ|g∈Γ

|Φ f (g)|,

by applying the Burnside lemma. Since Γ is fixed-free on X α,β by Lemma 6.1.4, we also

know that

ε( M /Γ) =ε( M )

|Γ| .




Applying the Euler-Poincare formula for the quotient map M /Γ, we get thatg∈Γ

|Φv(g)|

|Γ

|− ε( M )

|Γ

|+

g∈Γ

|Φ f (g)|

|Γ

|= χ( M /Γ).

Whence, g∈Γ

|Φv(g)| − ε( M ) +g∈Γ

|Φ f (g)| = |Γ| χ( M /Γ).

Notice that ν( M ) = |Φv(1Γ)|, φ( M ) = |Φ f (1Γ)| and ν( M ) − ε( M ) + φ( M ) = χ( M ). We find

that

χ( M ) +

g∈Γ,g1Γ

(|Φv(g)| + |Φ f (g)|) = |Γ| χ( M /Γ).

Furthermore, if Γ is fixed-free on V ( M ), by Theorem 6.2.5 there is a voltage assign-

ment ϑ on the quotient map M /Γ such that M ( M /G)

ϑ

. According to Theorem 6.1.7,there must be

ν( M /Γ) =ν( M )

|Γ| .

Whence,

g∈Γ

|Φv(g)| = ν( M ) and

g∈Γ,g1Γ

(|Φv(g)| = 0. Therefore, we get that

χ( M ) +

g∈Γ,g1Γ

|Φ f (g)| = |Γ| χ( M /Γ).

Consider the action properties of group Γ on F ( M ), we immediately get some inter-

esting results following.

Corollary 6.2.5 If Γ ≤ Aut M is fixed-free on V ( M ) and transitive on F ( M ) , for example,

M is regular and Γ = Aut M, then M /Γ is an one face map and

χ( M ) = |Γ|( χ( M /Γ) − 1) + φ( M ).

Corollary 6.2.6 For an one face map M, if Γ ≤ Aut M is fixed-free on V ( M ) , then

χ( M ) − 1 = |Γ|( χ( M /Γ) − 1),

and |Γ|. Particularly, |Aut M | is an integer factor of χ( M ) − 1.

Remark 6.2.1 For a one face planar map, i.e., the plane tree, the only fixed-free auto-

morphism group on its vertices is the trivial group by the Corollary 6.2.6.



Sec.6.3 Measures on Maps 227

§6.3 MEASURES ON MAPS

On the classical geometry, a central question is to determine the measures on objects,

such as those of the distance, angle, area, volume, curvature, . . .. For maps being that of

a combinatorial model of Klein surfaces, we also wish to introduce various measures on

maps and then enlarge its application to more branches of mathematics.

6.3.1 Angle on Map. For a map M = (X α,β, P), x ∈ X α,β, the permutation pair

( x, P x), (α x, P−1α x) is called an angle of M incident with x introduced by Tutte in

[Tut1]. We prove that any automorphism of a map is a conformal mapping and affirm the

Theorem 5.3.12 in Chapter 5 again in this section.

We define the angle transformation Θ of a map M = (X α,β, P) by

Θ = x∈X α,β

( x, P x).

Then we have

Theorem 6.3.1 Any automorphism of map M = (X α,β, P) is conformal.

Proof By the definition, for ∀g ∈ Aut M we know that

αg = gα, βg = g β and Pg = gP .

Therefore, for ∀ x ∈ X α,β,Θg( x) = (g( x), Pg( x))

and

gΘ( x) = g( x, P x) = (g( x), Pg( x)).

Whence, we get that for ∀ x ∈ X α,β, Θg( x) = gΘ( x). So Θg = gΘ,i.e., gΘg−1 = Θ.

Since for ∀ x ∈ X α,β, gΘg−1( x) = (g( x), Pg( x)) and Θ( x) = ( x, P( x)), we get that

(g( x), Pg( x)) = ( x, P( x)).

Thus g is a conformal mapping.

6.3.2 Non-Euclid Area on Map. For a voltage map ( M , σ) with a assignment σ :

X α,β( M ) → Γ, its non-Euclid area µ( M , Γ) is defined by

µ( M , Γ) = 2π(− χ( M ) +

m∈O(F ( M ))

(−1 +1

m)).




Particularly, since any map M can be viewed as a voltage map ( M , 1Γ), we get the non-

Euclid area of a map M

µ( M ) = µ( M , 1Γ) = −2πχ( M ).

Notice that the area of a map is only dependent on the genus of the surface. We know

the following result.

Theorem 6.3.2 Two maps on one surface S have the same non-Euclid area.

By the non-Euclid area, we find the Riemann-Hurwitz formula for map in the fol-

lowing.

Theorem 6.3.3 If Γ ≤ Aut M is fixed-free on V ( M ) , then

|Γ| = µ( M ) µ( M /Γ, ϑ)

,

where ϑ is constructed in the proof of the Theorem 6.2.5.

Proof According to the Theorem 6.2.6, we know that

|Γ| =− χ( M )

− χ( M ) +

m∈O(F ( M ))

(−1 + 1m

)

=−2πχ( M )

2π(− χ( M ) + m∈O(F ( M ))

(−1 +1m ))

=µ( M )

µ( M /Γ, ϑ)

.

As an interesting result, we can obtain the same result for the non-Euclid area of a

triangle as in the classical diff erential geometry following, seeing [Car1] for details.

Theorem 6.3.4 The non-Euclid area µ(∆) of a triangle ∆ on surface S with internal

angles η ,θ ,σ is

µ(∆) = η + θ + σ − π.

Proof According to the Theorems 4.2.1 and 6.2.5, we can assume that there exists

a triangulation M with internal angles η ,θ ,σ on S , and with an equal non-Euclid area on

each triangular disk. Then

φ( M ) µ(∆) = µ( M ) = −2πχ( M )

= −2π(ν( M ) − ε( M ) + φ( M )).



Sec.6.4 A Combinatorial Refinement of Huriwtz Theorem 229

Since M is a triangulation, we know that 2ε( M ) = 3φ( M ). Notice that the sum of all the

angles in the triangles on the surface S is 2πν( M ). We get that

φ( M ) µ(∆) =

−2π(ν( M )

−ε( M ) + φ( M )) = (2ν( M )

−φ( M ))π

=φ( M )i=1

[(η + θ + σ) − π] = φ( M )(η + θ + σ − π).

Whence, µ(∆) = η + θ + σ − π.

§6.4 A COMBINATORIAL REFINEMENT OF HURIWTZ THEOREM

6.4.1 Combinatorially Huriwtz Theorem. In 1893, Hurwitz obtained a famous result

on orientation-preserving automorphism groups Aut+S of Riemann surfaces S ([BEGG1],

[FaK1] and [GrT1]) following:

For a Riemann surface S of genus g(S) ≥ 2 , Aut+S ≤ 84(g(S) − 1).

We have established the combinatorial model for Klein surfaces, especially, the Riemann

surfaces by maps. Then what is its combinatorial counterpart? What can we know the

bound for the automorphisms group of map?

For a given graph Γ, a graphical property P is defined to be a family of its subgraphs,

such as, regular subgraphs, circuits, trees, stars, wheels, · · ·. LetM =

(X α,β, P) be a map.Call a subset A of X α,β has the graphical property P if its underlying graph of possesses

property P. Denote by A(P, M ) the set of all the A subset with property P in the map M .

For refining the Huriwtz theorem, we get a general combinatorial result in the fol-

lowing.

Theorem 6.4.1 Let M = (X α,β, P) be a map. Then for ∀ H ≤ Aut M,

[|v H ||v ∈ V ( M )] | | H |and

| H | | | A||A(P, M )|,where £ [a, b, · · ·] denotes the least common multiple of a, b, · · ·.

Proof According to Theorem 2.1.1(3), for ∀v ∈ V ( M ), | H | = | H v||v H |. So |v H | | | H |.Whence,

[|v H ||v ∈ V ( M )] | | H |.




We have know that the action of H on X α,β is fixed-free by Theorem 5.3.5, i.e., ∀ x ∈ X α,β,

there must be | H x| = 1. We consider the action of the automorphism group H on A(P, M ).

Notice that if A ∈ A(P, M ), then for ∀g ∈ H , Ag) ∈ A(P, M ), i.e., A H ⊆ A(P, M ).

Thus the action of H on A(P, M ) is closed. Whence, we can classify the elements inA(P, M ) by H . For ∀ x, y ∈ A(P, M ), define x ∼ y if and only if there is an element

g, g ∈ H such that xg = y.

Since | H x| = 1, i.e., | x H | = | H |, each orbit of H action on X α,β has a same length | H |.By the previous discussion, the action of H on A(P, M ) is closed. Therefore, the length

of each orbit of H action on A(P, M ) is | H |. Notice that there are | A||A(P, M )| quadricells

in A(P, M ). We get that

| H | | | A||A(P, M )|.


Choose the property P to be tours with each edge appearing at most 2 in the map M .

Then we get the following results by the Theorem 6 .4.1.

Corollary 6.4.1 Let T r 2 be the set of tours with each edge appearing 2 times. Then for

H ≤ Aut M,

| H | | (l|T r 2|, l = |T | =|T |2

≥ 1, T ∈ T r 2, ).

Let T r 1 be the set of tours without repeat edges. Then

| H | | (2l|T r 1|, l = |T | =|T |2

≥ 1, T ∈ T r 1, ).

Particularly, denote by φ(i, j) the number of faces in M with facial length i and singular

edges j, then

| H | | ((2i − j)φ(i, j), i, j ≥ 1),

where,(a, b, · · ·) denotes the greatest common divisor of a, b, · · ·.Corollary 6.4.2 Let T be the set of trees in the map M. Then for H ≤ Aut M,

| H | | (2lt l, l ≥ 1),

where t l denotes the number of trees with l edges.

Corollary 6.4.3 Let vi be the number of vertices with valence i. Then for H ≤ Aut M,

| H | | (2ivi, i ≥ 1).



Sec.6.4 A Combinatorial Refinement of Huriwtz Theorem 231

6.4.2 Application to Klein Surface. Theorem 6.4.1 is a combinatorial refinement of the

Hurwitz theorem. Applying it, we can get the automorphism group of map as follows.

Theorem 6.4.2 Let M be an orientable map of genus g( M )

≥2 and Γ+

≤Aut+ M,

Γ ≤ Aut M. Then

|Γ+| ≤ 84(g( M ) − 1) and |Γ| ≤ 168(g( M ) − 1).

Proof Define the average vertex valence ν( M ) and the average face valence φ( M ) of

a map M by

ν( M ) =1

ν( M )

i≥1

iνi,

φ( M ) = 1φ( M )

j≥1

jφ j,

where,ν( M ),φ( M ),φ( M ) and φ j denote the number of vertices, faces, vertices of valence i

and faces of valence j, respectively. Then we know that ν( M )ν( M ) = φ( M )φ( M ) = 2ε( M ).

Whence, ν( M ) =2ε( M )

ν( M )and φ( M ) =

2ε( M )

φ( M ). According to the Euler formula, we have

that

ν( M ) − ε( M ) + φ( M ) = 2 − 2g( M ),

where,ε( M ), g( M ) denote the number of edges and genus of the map M . We get that

ε( M ) =2(g( M ) − 1)

(1 − 2

ν( M )− 2

φ( M ))

.

Choose the integers k = ⌈ν( M )⌉ and l = ⌈φ( M )⌉. We find that

ε( M ) ≤ 2(g( M ) − 1)

(1 − 2k − 2

l)

.

Because of 1 − 2

k − 2

l> 0, So k ≥ 3, l >

2k

k − 2. Calculation shows that the minimum

value of 1

−2

k −2

l

is1

21

and attains the minimum value if and only if (k , l) = (3, 7) or

(7, 3). Therefore,

ε( M ≤ 42(g( M ) − 1)).

According to the Theorem 6.4.1 and its corollaries, we know that |Γ| ≤ 4ε( M ) and if

Γ+ is orientation-preserving, then |Γ+| ≤ 2ε( M ). Whence,

|Γ| ≤ 168(g( M ) − 1))




and

|Γ+| ≤ 84(g( M ) − 1)),

with equality hold if and only if Γ = Γ+ = AutM, (k , l) = (3, 7) or (7, 3).

For the automorphism of Riemann surface, we have

Corollary 6.4.4 For any Riemann surface S of genus g ≥ 2 ,

4g(S) + 2 ≤ |Aut+S| ≤ 84(g(S) − 1)

and

8g(S) + 4 ≤ |AutS| ≤ 168(g(S) − 1).

Proof By the Theorems 5.3.11 and 6.4.2, we know the upper bound for

|Aut

S|and

|Aut+S|. Now we prove the lower bound. We construct a regular map M k = (X k , Pk ) on

a Riemann surface of genus g ≥ 2 as follows, where k = 2g + 1.

X k = x1, x2, · · · , xk , α x1, α x2, · · · , α xk , β x1, β x2, · · · , β xk , αβ x1, αβ x2, · · · , αβ xk

Pk = ( x1, x2, · · · , xk , αβ x1, αβ x2, · · · , αβ xk )( β xk , · · · , β x2, β x1, α xk , · · · , α x2, α x1).

It can be shown that M k is a regular map, and its orientation-preserving automorphism

group Aut+ M k =< Pk >. Calculation shows that if k ≡ 0(mod 2), M k has 2 faces, and if

k ≡ 1, M k is an one face map. Therefore, By Theorem 5.3.11, we get that

|Aut+S| ≥ 2ε( M k ) ≥ 4g + 2,

and

|AutS| ≥ 4ε( M k ) ≥ 8g + 4.

For the non-orientable case, we can also get the bound for the automorphism group

of a map.

Theorem 6.4.3 Let M be a non-orientable map of genus g′( M ) ≥ 3. Then for Γ+ ≤Aut+ M,

|Γ+| ≤ 42(g′( M ) − 2)

and for Γ ≤ Aut M,

|Γ| ≤ 84(g′( M ) − 2),






surface without boundary of genus q ≥ 3. Their approach is by using the Fuchsian groups

or NEC groups for Klein surfaces. Their approach is by applying the Riemann-Hurwitz

equation, i.e., Theorem 4.4.5. Here we restate it in the following:

Let Γ be an NEC graph and Γ′ a subgroup of Γ with finite index. Then

µ(Γ′) µ(Γ)

= [Γ : Γ′],

where, µ(Γ) is the non-Euclid area of group Γ defined by

µ(G) = 2π[ηg + k − 2 +

r i=1

(1 − 1/mi) + 1/2

k i=1

si j=1

(1 − 1/ni j)]

if the signature of the group Γ is

σ = (g; ±; [m1, · · · , mr ]; (n11,···,n1s1), · · · , (nk 1, · · · , nks )),

where, η = 2 if sign(σ) = + and η = 1 otherwise.

Notice that we have introduced the conception of non-Euclid area for the voltage

maps and have gotten the Riemann-Hurwitz equation in Theorem 6.2.6 for a group action

fixed-free on vertices of map. Similarly, we can find the minimum genus of a fixed-free

automorphism of a map on its vertex set by the voltage assignment technique on one of

its quotient map and get the maximum order of an automorphism of map.

Lemma 6.5.1 Let N =k

i=1 p

r ii

, p1 < p2 < · · · < pk be the arithmetic decomposition of an

integer N and mi ≥ 1, mi| N for i = 1, 2, · · · , k. Then for any integer s ≥ 1 ,

si=1

(1 − 1

mi

) ≥ 2(1 − 1

p1

)⌊ s

2⌋.

Proof If s ≡ 0(mod 2), it is obvious that

s

i=1

(1 −1

mi ) ≥s

i=1

(1 −1

p1 ) ≥ (1 −1

p1 )s.

Assume that s ≡ 1(mod 2) and there are mi j p1, j = 1, 2, · · · , l. If the assertion is not

true, we must have that

(1 − 1

p1

)(l − 1) >

l j=1

(1 − 1

mi j

) ≥ (1 − 1

p2

)l.



Sec.6.5 The Order of Automorphism of Klein Surface 235

Whence,

(1 − 1

p1

)l > (1 − 1

p2

)l + 1 − 1

p1

> (1 − 1

p1

)l,

a contradiction. Therefore, we get that

si=1

(1− 1

mi

) ≥ 2(1− 1

p1

)⌊ s

2⌋.

Lemma 6.5.2 For a map M = (X α,β, P) with φ( M ) faces and N =k

i=1

pr ii

, p1 < p2 <

· · · < pk , the arithmetic decomposition of an integer N , there exists a voltage assignment

ϑ : X α,β → Z N such that for ∀F ∈ F ( M ) , o(F ) = p1 if φ( M ) ≡ 0(mod 2) or there exists a

face F 0 ∈ F ( M ) such that o(F ) = p1 for ∀F ∈ F ( M ) \ F 0 , but o(F 0) = 1.

Proof Assume that f 1, f 2, · · · , f n are the n faces of the map M , where n = φ( M ). By

the definition of voltage assignment, if x, β x or x, αβ x appear on one face f i, 1 ≤ i ≤ n

altogether, then they contribute to ϑ( f i) only with ϑ( x)ϑ−1( x) = 1 Z N . Whence, not loss of

generality, we only need to consider the voltage xi j on the common boundary among the

faces f i and f j for 1 ≤ i, j ≤ n. Then the voltage assignment on the n faces are

ϑ( f 1) = x12 x13 · · · x1n,

ϑ( f 2) = x21 x23

· · · x2n,

· · · · · · · · · · · · · · · · · ·

ϑ( f n) = xn1 xn2 · · · xn(n−1).

We wish to find an assignment on M which can enables us to get as many faces as possible

with the voltage of order p1. Not loss of generality, we choose ϑ p1 ( f 1) = 1 Z N in the first.

To make ϑ p1 ( f 2) = 1 Z N , choose x23 = x−1

13, · · · , x2n = x−1

1n. If we have gotten ϑ p1 ( f i) = 1 Z N

and i < n if n ≡ 0(mod 2) or i < n − 1 if n ≡ 1(mod 2), we can choose that

x(i+1)(i+2) = x−1i(i+2), x(i+1)(i+3) = x−1

i(i+3), · · · , x(i+1)n = x−1in ,

which also make ϑ p1 ( f i+1) = 1 Z N .

Now if n ≡ 0(mod 2), this voltage assignment makes each face f i, 1 ≤ i ≤ n satisfying

that ϑ p1 ( f i) = 1 Z N . But if n ≡ 1(mod 2), it only makes ϑ p1 ( f i) = 1 Z N

for 1 ≤ i ≤ n − 1, but

ϑ( f n) = 1 Z N . This completes the proof.




Now we can find a result on the minimum genus of a fixed-free automorphism of

map by Lemmas 6.5.1-6.5.2 following.

Theorem 6.5.1 Let M = (X α,β, P) be a map and N = pr 11

· · · p

r k

k , p1 < p2 <

· · ·< pk the

arithmetic decomposition of integer N. Then for any voltage assignment ϑ : X α,β → Z N ,

(1) If M is orientable, the minimum genus gmin of the lifted map M ϑ which admits a

fixed-free automorphism on V ( M ϑ) of order N is

gmin = 1 + N g( M ) − 1 + (1 −

m∈O(F ( M ))

1

p1

)⌊φ( M )

2⌋.

(2) If M is non-orientable, the minimum genus g′min

of the lifted map M ϑ which

admits a fixed-free automorphism on V ( M ϑ) of order N is

g′min = 2 + N g( M ) − 2 + 2(1 − 1

p1

)⌊φ( M )

2⌋.

Proof (1) According to Theorem 6.2.5, we know that

2 − 2g( M ϑ) = N (2 − 2g( M )) +

m∈O(F ( M ))

(−1 +1

m).

Whence,

2g( M ϑ) = 2 + N 2g( M )

−2 + m∈O(F ( M ))

(1−

1

m).

Applying Lemmas 6.5.1 and 6.5.2, we get that

gmin = 1 + N g( M ) − 1 + (1 − 1

p1

)⌊φ( M )

2⌋

. (2) Similarly, by Theorem 6.2.1, we know that

2 − g( M ϑ) = N (2 − g( M )) +

m∈O(F ( M ))

(−1 +1

m).

Whence,

g( M ϑ) = 2 + N g( M ) − 2 +

m∈O(F ( M ))

(1 − 1

m).

Applying Lemmas 6.5.1 and 6.5.2, we get that

g′min = 2 + N g( M ) − 2 + 2(1 − 1

p1

)⌊φ( M )

2⌋.



Sec.6.5 The Order of Automorphism of Klein Surface 237

6.5.2 The Maximum Order of Automorphisms of a Map. For the maximum order of

automorphisms of a map, we have the following result.

Theorem 6.5.2 The maximum order N max of automorphisms g of an orientable map M

with genus≥ 2 is

N max ≤ 2g( M ) + 1

and the maximum order N ′max of automorphisms g of a non-orientable map with genus≥ 3

is

N ′max ≤ g( M ) + 1,

where g( M ) denotes the genus of map M.

Proof According to Theorem 6.2.3, denote by Γ = g, we get that

χ( M ) +

g∈Γ,g1Γ

(|Φv(g)| + |Φ f (g)|) = |Γ| χ( M /Γ),

where, Φ f (g) = F |F ∈ F ( M ), F g = F and Φv(g) = v|v ∈ V ( M ), vg = v. Notice

that a vertex of M is a pair of conjugacy cycles in P , and a face of M is a pair of

conjugacy cycles in Pαβ. If g 1Γ, direct calculation shows that Φ f (g) = Φ f (g2) and

Φv(g) = Φv(g2). Whence,

g∈Γ,g1Γ

|Φv(g)| = (|Γ| − 1)|Φv(g)|

and

g∈Γ,g1Γ

|Φ f (g)| = (|Γ| − 1)|Φ f (g)|.

Therefore, we get that

χ( M ) + (|Γ| − 1)|Φv(g)| + (|Γ| − 1)|Φ f (g)| = |Γ| χ( M /Γ).

Whence,

χ( M ) − (|Φv(g)| + |Φ f (g)|) = |Γ|( χ( M /Γ) − (|Φv(g)| + |Φ f (g)|)).




If χ( M /G) − (|Φv(g)| + |Φ f (g)|) = 0, i.e., χ( M /Γ) = |Φv(g)| + |Φ f (g)| ≥ 0, then we get

that g( M ) ≤ 1 if M is orientable or g( M ) ≤ 2 if M is non-orientable. Contradicts to the

assumption. Therefore, χ( M /Γ) − (|Φv(g)| + |Φ f (g)|) 0. Whence, we get that

|Γ| = χ( M ) − (|Φv(g)| + |Φ f (g)|)

χ( M /Γ) − (|Φv(g)| + |Φ f (g)|) = H (v, f ; g).

Notice that |Γ|, χ( M )− (|Φv(g)| + |Φ f (g)|) and χ( M /G)− (|Φv(g)| + |Φ f (g)|) are integers. We

know that the function H (v, f ; g) takes its maximum value at χ( M /Γ)−(|Φv(g)|+|Φ f (g)|) =

−1 since χ( M ) ≤ −1. But in this case, we get that

|Γ| = |Φv(g)| + |Φ f (g)| − χ( M ) = 1 + χ( M /Γ) − χ( M ).

We divide our discussion into two cases.

Case 1. M is orientable.

Since χ( M /Γ) + 1 = (|Φv(g)| + |Φ f (g)|) ≥ 0, we know that χ( M /Γ) ≥ −1. Whence,

χ( M /Γ) = 0 or 2. We get that

|Γ| = 1 + χ( M /Γ) − χ( M ) ≤ 3 − χ( M ) = 2g( M ) + 1.

That is, N max

≤2g( M ) + 1.

Case 2. M is non-orientable.

In this case, since χ( M /Γ) ≥ −1, we know that χ( M /Γ) = −1, 0, 1 or 2. Whence,

we get that

|Γ| = 1 + χ( M /Γ) − χ( M ) ≤ 3 − χ( M ) = g( M ) + 1.


According to this theorem, we get the following result for the order of an automor-

phism of a Klein surface without boundary by the Theorem 5.3.12, which is even more

better than the results already known.

Corollary 6.5.1 The maximum order of conformal transformations realizable by maps M

on a Riemann surface of genus≥ 2 is 2g( M ) + 1 and the maximum order of conformal

transformations realizable by maps M on a non-orientable Klein surface of genus≥ 3

without boundary is g( M ) + 1.



Sec.6.6 Remarks 239

The maximum order of an automorphism of map can be also determined by its un-

derlying graph as follows.

Theorem 6.5.3 Let M be a map underlying graph G and let omax( M , g), omax(G, g) be the

maximum orders of orientation-preserving automorphisms in Aut M and in Aut 12

G. Then

omax( M , g) ≤ omax(G, g),

and the equality holds for at least one such map M underlying graph G.

The proof of the Theorem 6.5.3 will be delayed to the next chapter after we proved

Theorem 7.1.1. By this result, we find some interesting conclusions following.

Corollary 6.5.2 The maximum order of orientation-preserving automorphisms of a com-

plete map K n, n ≥ 3 is at most n.

Corollary 6.5.3 The maximum order of orientation-preserving automorphisms of a plane

tree T is at most |T |− 1 and attains the upper bound only if the underlying tree is a star.

§6.6 REMARKS

6.6.1 The lifted graph of a voltage graph (G, σ) with σ : X 12

(G)

→Γ is in fact a regular

covering of 1-complex G constructing dependent on a group (Γ; ). This technique was

extensively applied to coloring problem, particularly, its dual, i.e., current graph for deter-

mining the genus of complete graph K n on surface. The reference [GrT1] is an excellent

book systematically dealing with voltage graphs. One can also find the combinatorial

counterparts of a few important results, such as those of the Riemann-Hurwitz equation

and Alexander’s theorem on branch points in Riemann geometry in this book. Certainly,

the references [Liu1] and [Whi1] also partially discuss voltage graphs. A similar consid-

eration for non-regular covering space presents the following problem:

Problem 6.6.1 Apply the voltage assignment technique for constructing non-regular cov-

ering of graphs or maps.

6.6.2 The technique of voltage graphs and voltage maps is essentially a discrete realiza-

tion of regular covering spaces with dimensional 1 or 2. Many results on covering spaces

can be found the combinatorial counterparts in voltage graphs or maps. For example,




Theorem 6.1.1 asserts that if π : S → S is a covering projection, then for any arc f in S

with initial point x0 there exists a unique lifting arc f l with initial point

x0 in S . In voltage

graphs, we know its combinatorial counterpart following.

Theorem 6.6.1 Let W be a walk with initial vertex u ∈ V (G) in a voltage graph (G, σ)

with assignment σ : X 12

(G) → Γ and g ∈ Γ. then there is a unique lifting of W that starts

at ug in Gσ.

Certainly, there are many such results by finding the combinatorial counterparts, for

example in voltage graphs or maps for results known in topology or geometry. The book

[MoT1] can be seen as a discrete deal with surface geometry, i.e., combinatorics on sur-

face geometry. These results in Sections 4 and 5 are also such kind results. Generally, a

combinatorial speculation for mathematical science will finally arrived at the CC conjec-

ture for developing mathematics discussed in the final chapter of this book.

6.6.3 For a map ( M , σ) with voltage assignment σ : X α,β( M ) → Γ, it is easily to know

that the group (Γ; ) is a map group of M σ action closed in each fiber π−1( x) for x ∈X α,β( M ), i.e., Γ ≤ Aut M σ. In this way, one can get regular maps in lifted maps. Such a

role of voltage maps is known in Theorem 6.2.2, which enables one to get regular maps by

voltage assignments. Similarly, the exponent group Ex( M ) of map and the construction

of derived map M σ,ι also enables one to find more regular maps. The reader is refereed to

[Ned1] and [NeS1] for its techniques.

6.6.4 Theorem 6.2.5 is an important result related the quotient map with that of voltage

assignment, which enables one to find relations between voltage group, Euler-Poincare

characteristic and fixed point sets. Theorems 6.2.6 and 6.2.7 are such results. This theo-

rem is in fact a generalization of a result on voltage graph following, obtained by Gross

and Tucker in 1974.

Theorem 6.6.2 Let A be a group acting freely on a graph

G and let G be the resulting

quotient graph. Then there is an assignment σ of voltages inA

to the quotient graph G

and a labeling of the vertices of G by the elements of V (G) × A such that G = Gσ and

that the given action of A on G is the natural left action of A on Gσ.

6.6.5 For applying ideas of maps to metric mathematics, various metrics on maps are need

to introduce besides angles and non-Euclid area discussed in Section 3. For example,

the length and arc length, the circumference, the volume and the curvature, · · ·, which



Sec.6.6 Remarks 241

needs one to speculate the classical mathematics by combinatorics, i.e., combinatorially

reconstruct such a mathematical science.

6.6.6 We have know that maps can be viewed as a combinatorial model of Klein surfaces

in Chapter 5. Usually, a problem is difficult in Klein surface but it is easy for its counter-

part in combinatorics, such as those in Corollary 6.5.1. Further applying this need us to

solve the following problem.

Problem 6.6.2 Determine these behaviors of Klein surfaces S , such as automorphisms

that can not be realizable by maps M on S .

As we known, there are few results on Problem 6 .6.1 in publication. But it is funda-

mental for applying combinatorial technique to metric mathematics.



CHAPTER 7.

Map Automorphisms Underlying a Graph

A complete classification of non-equivalent embeddings of graph G on sur-faces or maps M = (X α,β, P) underlying G requires to find permutation

presentations of automorphisms of G on X α,β. For this objective, an alter-

nate approach is to consider the induced action of semi-arc automorphisms

of graph G( M ) on quadricells X α,β. In fact, the automorphism group Aut M

is nothing but consisting of all such automorphisms g|X α,β that Pg|X α,β= P .

Topics covered in this chapter include a necessary and sufficient characteris-

tic for a subgroup of G being that of map and permutation presentations for

automorphisms of maps underlying a complete graph, a semi-regular graph

or a bouquet. Certainly, these presentations of complete maps or semi-regular

maps can be also applied to maps underlying wheels K 1 + C n or GRR graphs

of a finite group (Γ; ). All of these permutation presentations are typical ex-

amples for characterizing the behavior of map groups, and can be also applied

for the enumeration of non-isomorphic maps in Chapter 8.



Sec.7.1 A Condition for Graph Group Being That of Map 243

§7.1 A CONDITION FOR GRAPH GROUP BEING THAT OF MAP

7.1.1 Orientation-Preserving or Reversing. Let G = (V , E ) be a connected graph.

Its automorphism is denoted by AutG. Choose the base set of maps underlying G to be

X = E . Then its quadricells X α,β is defined by

X α,β = x∈ X

x, α x, β x, βαβ x,

where, K = 1,α,β,αβ is the Klein 4-elements group. For ∀g ∈ AutG, an induced action

g|X α,β of g on X α,β is defined as follows:

For ∀ x ∈ X α,β , if xg = y, then define (α x)g = α y, ( β x)g = β y and (αβ x)g = αβ y.

Let M = (X α,β, P) be a map. According to the Theorem 5.3.8, for an automorphism g ∈Aut M , let g|V ( M ) : u → v, u, v ∈ V ( M ). If ug = v, then g is called an orientation-preserving

automorphism and if ug = v−1, such a g is called an orientation-reversing automorphism.

For any g ∈ Aut M , it is obvious that g|G is orientation-preserving or orientation-reversing,

and the product of two orientation-preserving or orientation-reversing automorphisms is

orientation-preserving, but the product of an orientation-preserving with an orientation-

reversing automorphism is orientation-reversing.

For a subgroup Γ ≤ Aut M , define Γ+ ≤ Γ being the orientation-preserving sub-

group of H . Then it is clear that the index of Γ

+

in Γ is 2. Let v be a vertex withv = ( x1, x2, · · · , x ρ(v))(α x ρ(v), · · · , α x2, α x1). Denote by v the cyclic group generated by v.

Then we get a property following for automorphisms of a map.

Lemma 7.1.1 Let Γ ≤ Aut M be an automorphism group of map M. Then ∀v ∈ V ( M ) ,

(1) If ∀g ∈ Γ , g is orientation-preserving, then Γv ≤ v is a cyclic group;

(2) Γv ≤ v × α.

Proof (i) Let M = (X α,β, P). For any ∀g ∈ G, since g is orientation-preserving, we

know that vh

= v for ∀v ∈ V ( M ), h ∈ Γv. Assume

v = ( x1, x2, · · · , x ρ(v))(α x ρ(v), α x ρ(v)−1, · · · , α x1).

Then

[( x1, x2, · · · , x ρ(v))(α x ρ(v), · · · , α x2, α x1)]h = ( x1, x2, · · · , x ρ(v))(α x ρ(v), · · · , α x2, α x1).



244 Chap.7 Map Automorphisms Underlying a Graph

Therefore, if h( x1) = xk +1, 1 ≤ k ≤ ρ(v), then

h = [( x1, x2, · · · , x ρ(v))(α x ρ(v), α x ρ(v)−1, · · · , α x1)]k = vk .

Now if h( x1) = α x ρ(v)−k +1, 1 ≤ k ≤ ρ(v), then

h = [( x1, x2, · · · , x ρ(v))(α x ρ(v), α x ρ(v)−1, · · · , α x1)]k α = vk α.

But if h = vk α, we know that vh = vα = v−1, i.e., h is not orientation-preserving. Whence,

h = vk , 1 ≤ k ≤ ρ(v), i.e., every element in Γv is a power of v. Let ξ be the least power of

elements in Γv. Then Γv =

v ξ

≤ v is a cyclic group generated by v ξ .

(2) For ∀g ∈ Gv, vg = v, i.e.,

[( x1, x2, · · · , x ρ)(α x ρ, α x ρ−1, · · · , α x1)]g = ( x1, x2, · · · , x ρ)(α x ρ, α x ρ−1, · · · , α x1).

Similar to the proof of (1), we know that there exists an integer s, 1 ≤ s ≤ ρ such that

g = vs or g = vsα. Consequently, g ∈ v or g ∈ v α, i.e.,

Γv ≤ v × α .

Lemma 7.1.2 Let G be a connected graph. If Γ ≤ AutΓ , and ∀v ∈ V (G) , Γv ≤ v × α ,then the action of Γ on X α,β is fixed-free.

Proof Choose a quadricell x ∈ X α,β. We prove that Γ x = 1X α,β. In fact, if g ∈ Γ x,

then xg = x. Particularly, the incident vertex u is stable under the action of g, i.e., ug = u.

Let

u = ( x, y1, · · · , y ρ(u)−1)(α x, α y ρ(u)−1, · · · , α y1),

then because of Γu ≤ u × α, we get that

xg = x, yg

1= y1, · · · , y

g

ρ(u)−1= y ρ(u)−1

and

(α x)g = α x, (α y1)g = α y1, · · · , (α y ρ(u)−1)g = α y ρ(u)−1,

thus for any quadricell eu incident with the vertex u, egu = eu. According to the definition

of induced action AutG on X α,β, we know that

( β x)g = β x, ( β y1)g = β y1, · · · , ( β y ρ(u)−1)g = β y ρ(u)−1




and

(αβ x)g = αβ x, (αβ y1)g = αβ y1, · · · , (αβ y ρ(u)−1)g = αβ y ρ(u)−1.

Whence, for any quadricell y

∈X α,β, if the incident vertex of y is w, then by the connect-

edness of graph G, there is a path P(u, w) = uv1v2 · · · vsw connecting the vertices u and

w in G. Not loss of generality, we assume that β yk is incident with the vertex v1. Since

( β yk )g = β yk and Γv1

≤ v1 × α, we know that for any quadricell ev1incident with the

vertex v1, egv1

= ev1.

Similarly, if a quadricell eviincident with the vertex vi is stable under the action of g,

i.e., (evi)g = evi

, then we can prove that any quadricell evi+1incident with the vertex vi+1 is

stable under the action of g. This process can be well done until we arrive the vertex w.

Therefore, we know that any quadricell ew incident with the vertex w is stable under the

action of g. Particularly, we get that yg = y.

Therefore, g = 1Γ. Whence, Γ x = 1Γ.

7.1.2 Group of a Graph Being That of Map. Now we obtain a necessary and sufficient

condition for a subgroup of a graph being that an automorphism group of map underlying

this graph.

Theorem 7.1.1 Let G be a connected graph. If Γ ≤ AutG, then Γ is an automorphism

group of map underlying graph G if and only if for

∀v

∈V (G) , the stabilizer Γv

≤ v

×α

.

Proof According to Lemma 7.1.1(ii), the condition of Theorem 7.1.1 is necessary.

Now we prove its sufficiency.

By Lemma 7.1.2, we know that the action of Γ on X α,β is fixed-free, i.e., for∀ x ∈X α,β, |Γ x| = 1X α,β

. Whence, the length of orbit of x under the action of Γ is | xΓ| = |Γ x|| xΓ| =

|Γ|, i.e., for ∀ x ∈ X α,β, the length of orbit of x under the action of Γ is |Γ|.Assume that there are s orbits O1, O2, · · · , Os in V (Γ) under the action of Γ, where,

O1 = u1, u2, · · · , uk ,

O2 = v1, v2, · · · , vl,· · · · · · · · · · · · · · · · · ·,Os = w1, w2, · · · , wt .

We construct a conjugatcy permutation pair for every vertex in the graph G such that their

product P is stable under the action of Γ.

Notice that for ∀u ∈ V (G), because of |Γ| = |Γu||uΓ|, we know that [k , l, · · · , t ] | |Γ|.




First, we determine the conjugatcy permutation pairs for each vertex in the orbit O1.

Choose any vertex u1 ∈ O1. Assume that the stabilizer Γu1is 1X α,β

, g1, g2g1, · · · ,m−1i=1

gm−i,

where, m =

|Γu1

|and the quadricells incident with vertex u1 is N (u1) in the graph G. We

arrange the elements in N (u1) as follows.

Choose a quadricell ua1

∈ N (u1). We apply Γu1action on ua

1 and αua1, respectively.

Then we get a quadricell set A1 = ua1

, g1(ua1), · · · ,

m−1i=1

gm−i(ua1) and α A1 = αua

1, αg1(ua

1), · · · ,

αm−1i=1

gm−i(ua1). By the definition of a graph automorphism action on its quadricells, we

know that A1

α A1 = ∅. Arrange the elements in A1 as

−→ A1 = ua

1, g1(ua1), · · · ,

m−1i=1

gm−i(ua1).

If N (u1) \ A1

α A1 = ∅, then the arrangement of elements in N (u1) is

−→ A1. If

N (u1) \ A1α A1 ∅, choose a quadricell u

b

1 ∈ N (u1) \ A1 α A1. Similarly, apply-ing the group Γu1

acts on ub1, we get that A2 = ub

1, g1(ub1), · · · ,

m−1i=1

gm−i(ub1) and α A2 =

αub1

, αg1(ub1), · · · , α

m−1i=1

gm−i(ub1). Arrange the elements in A1

A2 as

−−−−−−−→ A1

A2 = ua

1, g1(ua1), · · · ,

m−1i=1

gm−i(ua1); ub

1, g1(ub1), · · · ,

m−1i=1

gm−i(ub1).

If N (u1)\( A1

A2

α A1

α A2) = ∅, then the arrangement of elements in A1

A2 is

−−−−−−−→ A1 A2. Otherwise, N (u1)\( A1 A2α A1α A2) ∅. We can choose another quadri-

cell uc1∈ N (u1) \ ( A1

A2

α A1

α A2). Generally, If we have gotten the quadricell sets

A1, A2, · · · , Ar , 1 ≤ r ≤ 2k , and the arrangement of element in them is−−−−−−−−−−−−−−−−−−−−→ A1

A2

· · ·

Ar ,

if N (u1) \ ( A1

A2

· · · Ar

α A1

α A2

· · ·α Ar ) ∅, we can choose an element

ud 1

∈ N (u1) \ ( A1

A2

· · · Ar

α A1

α A2

· · ·α Ar ) and define the quadricell set

Ar +1 = ud 1, g1(ud

1), · · · ,

m−1i=1

gm−i(ud 1)

α Ar +1 = αud 1, αg1(ud

1), · · · , α

m−1i=1

gm−i(ud 1)

and the arrangement of elements in Ar +1 is

−−−→ Ar +1 = ud

1, g1(ud 1), · · · ,

m−1i=1

gm−i(ud 1).




Now define the arrangement of elements inr +1 j=1

A j to be

−−−−→r +1 j=1

A j =

−−−−→r

i=1 Ai; −−−→ Ar +1.

Whence,

N (u1) = (

k j=1

A j)

(α

k j=1

A j)

and Ak is obtained by the action of the stabilizer Γu1on ue

1. At the same time, the arrange-

ment of elements in the subsetk

j=1 A j of N (u1) to be

−−−−→k

j=1

A j.

We define the conjugatcy permutation pair of the vertex u1 to be

u1

= (C )(αC −1α),

where £

C = (ua1, ub

1, · · · , ue1; g1(ua

1), g1(ub1), · · · , g1(ue

1), · · · ,

m−1i=1

(ua1),

m−1i=1

(ub1), · · · ,

m−1i=1

(ue1)).

For any vertex ui ∈ O1, 1 ≤ i ≤ k , assume that h(u1) = ui, where h ∈ G, we define the

conjugatcy permutation pair ui

of the vertex ui

to be

ui

= hu1

= (C h)(αC −1α−1).

Since O1 is an orbit of the action G on V (Γ), then we get that

(

k i=1

ui

)Γ =

k i=1

ui

.

Similarly, we can define the conjugatcy permutation pairs v1, v2

, · · · , vl, · · · , w1

,

w

2

,· · ·

, wt

of vertices in the orbits O2,· · ·

, Os. We also have that

(

li=1

vi

)Γ =

li=1

vi

.

· · · · · · · · · · · · · · · · · · · · ·

(

t i=1

wi

)Γ =

t i=1

wi

.




Now define the permutation

P = (

k

i=1

ui

) × (

l

i=1

vi

) × · · · × (

t

i=1

wi

).

Since all O1, O2, · · · , Os are the orbits of V (G) under the action of Γ, we get that

P Γ = (

k i=1

ui

)Γ × (

li=1

vi

)Γ × · · · × (

t i=1

wi

)Γ

= (

k i=1

ui

) × (

li=1

vi

) × · · · × (

t i=1

wi

) = P.

Whence, if let map M = (X α,β, P) £ then Γ is an automorphism of M .

For the orientation-preserving automorphisms, we know the following result.

Theorem 7.1.2 Let G be a connected graph. If Γ ≤ AutG, then Γ is an orientation-

preserving automorphism group of map underlying graph G if and only if for ∀v ∈ V (G) ,

the stabilizer Γv ≤ v is a cyclic group.

Proof According to Lemma 7.1.1(i) £ we know the necessary. Notice that the ap-

proach of construction the conjugatcy permutation pair in the proof of Theorem 7 .1.1 can

be also applied in the orientation-preserving case. We know that Γ is also an orientation-

preserving automorphism group of map M .

Corollary 7.1.1 For any positive integer n, there exists a vertex transitive map M un-

derlying a circultant such that Z n is an orientation-preserving automorphism group of

M.

By Theorem 7.1.2, we can prove the Theorem 6.5.3 now.

The Proof of Theorem 6.5.3

Since every subgroup of a cyclic group is also a cyclic group, we know that any cyclic

orientation-preserving automorphism group of the graph G is an orientation-preserving

automorphism group of a map underlying Γ by Theorem 7.1.2. Whence, we get that

omax( M , g) ≤ omax(G, g).

Note 7.1.1 Gardiner et al. proved in [GNSS1] that if add an additional condition in The-

orem 7.1.1, i.e, Γ is transitive on the vertices in G, then there is a regular map underlying

the graph G.



Sec.7.2 Automorphisms of a Complete Graph on Surfaces 249

§7.2 AUTOMORPHISMS OF A COMPLETE GRAPH ON SURFACES

7.2.1 Complete Map. A map is called a complete map if its underlying graph is a

complete graph. For a connected graph G, the notationsE

O(G),E

N (G) andE

L(G) denote

the embeddings of Γ on the orientable surfaces, non-orientable surfaces and locally sur-

faces, respectively. For ∀e = (u, v) ∈ E (G), its quadricell Ke = e, αe, βe, αβe can be

represented by Ke = uv+, uv−, vu+, vu−.

Let K n be a complete graph of order n. Label its vertices by integers 1, 2, · · · , n. Then

its edge set is i j|1 ≤ i, j ≤ n, i j i j = ji and

X α,β(K n) = i j+ : 1 ≤ i, j ≤ n, i j

i j− : 1 ≤ i, j ≤ n, i j,

α = 1≤i, j≤n,i j

(i j+, i j−),

β =

1≤i, j≤n,i j

(i j+, i j+)(i j−, i j−).

We determine all automorphisms of complete maps of order n and find presentations

for them in this section.

First, we need some useful lemmas for an automorphism of map induced by an

automorphism of its underlying graph.

Lemma 7.2.1 Let G be a connected graph and g

∈AutG. If there is a map M

∈ E L(G)

such that the induced action g∗ ∈ Aut M, then for ∀(u, v), ( x, y) ∈ E (G) ,

[lg(u), lg(v)] = [lg( x), lg( y)] = constant ,

where, lg(w) denotes the length of the cycle containing the vertex w in the cycle decompo-

sition of g.

Proof According to the Lemma 6.2.1, we know that the length of a quadricell uv+ or

uv− under the action g∗ is [lg(u), lg(v)]. Since g∗ is an automorphism of map, therefore, g∗

is semi-regular. Whence, we get that

[lg(u), lg(v)] = [lg( x), lg( y)] = constant.

Now we consider conditions for an induced automorphism of map by that of graph

to be an orientation-reversing automorphism of map.

Lemma 7.2.2 If ξα is an automorphism of map, then ξα = αξ.




Proof Since ξα is an automorphism of map, we know that

( ξα)α = α( ξα).

That is, ξα = αξ .

Lemma 7.2.3 If ξ is an automorphism of map M = (X α,β, P) , then ξα is semi-regular on

X α,β with order o( ξ ) if o( ξ ) ≡ 0(mod 2) and 2o( ξ ) if o( ξ ) ≡ 1(mod 2).

Proof Since ξ is an automorphism of map by Lemma 7.2.2, we know that the cyclic

decomposition of ξ can be represented by

ξ =

k

( x1, x2, · · · , xk )(α x1, α x2, · · · , α xk ),

where, k denotes the product of disjoint cycles with length k = o( ξ ).

Therefore, if k ≡ 0(mod 2), then

ξα =

k

( x1, α x2, x3, · · · , α xk )

and if k ≡ 1(mod 2), then

ξα =

2k

( x1, α x2, x3, · · · , xk , α x1, x2, α x3, · · · , α xk ).

Whence, ξ is semi-regular acting on X α,β.

Now we can prove the following result for orientation-reversing automorphisms of maps.

Lemma 7.2.4 For a connected graph G, let K be all automorphisms in AutG whose

extending action on X α,β , X = E (G) are automorphisms of maps underlying graph G.

Then for ∀ ξ ∈ K , o( ξ ∗) ≥ 2 , ξ ∗α ∈ K if and only if o( ξ ∗) ≡ 0(mod 2).

Proof Notice that by Lemma 7.2.3, if ξ ∗ is an automorphism of map underlying

graph G, then ξ ∗α is semi-regular acting on X α,β.

Assume ξ ∗ is an automorphism of map M = (X α,β, P). Without loss of generality,

we assume that

P = C 1C 2 · · · C k ,

where,C i = ( xi1, xi2, · · · , xi ji) is a cycle in the decomposition of ξ |V (G) and xit = (ei1, ei2,

· · · , eit i )(αei1, αeit i , · · · , αei2) and.

ξ | E (G) = (e11, e12, · · · , es1)(e21, e22, · · · , e2s2

) · · · (el1, el2, · · · , elsl).




and

ξ ∗ = C (αC −1α),

where, C = (e11, e12,

· · ·, es1

)(e21, e22,

· · ·, e2s2

)· · ·

(el1, el2,

· · ·, elsl

). Now since ξ ∗ is an

automorphism of map, we get that s1 = s2 = · · · = sl = o( ξ ∗) = s.

If o( ξ ∗) ≡ 0(mod 2), define a map M ∗ = (X α,β, P∗) with

P∗= C ∗1C ∗2 · · · C ∗k ,

where, C ∗i = ( x∗i1, x∗

i2, · · · , x∗i ji

), x∗it = (e∗

i1, e∗i2, · · · , e∗

it i)(αe∗

i1, αe∗it i

, · · · , e∗i2) and e∗

i j = e pq.

Take e∗i j = e pq if q ≡ 1(mod 2) and e∗

i j = αe pq if q ≡ 0(mod 2). Then we get that M ξα = M .

Now if o( ξ ∗) ≡ 1(mod 2), by Lemma 7.2.3, o( ξ ∗α) = 2o( ξ ∗). Therefore, any chosen

quadricells (ei1, ei2, · · · , eit i ) adjacent to the vertex xi1 for i = 1, 2, · · · , n, where, n = |G|, theresultant map M is unstable under the action of ξα. Whence, ξα is not an automorphism

of map underlying graph G.

7.2.2 Automorphisms of Complete Map. We determine all automorphisms of complete

maps of order n by applying the previous results. Recall that the automorphism group of

K n is the symmetry group of degree n, that is, AutK n = S V (K n).

Theorem 7.2.1 All orientation-preserving automorphisms of non-orientable complete

maps of order n ≥ 4 are extended actions of elements in

E[s

ns ]

, E[1,s

n−1s ]

,

and all orientation-reversing automorphisms of non-orientable complete maps of order

n ≥ 4 are extended actions of elements in

αE[(2s)

n2s ]

, αE[(2s)

42s ]

, αE[1,1,2],

where,Eθ

denotes the conjugatcy class containing element θ in the symmetry group of

degree n.

Proof First, we prove that an induced permutation ξ ∗ on a complete map of order

n by an element ξ ∈ S V (K n) is a cyclic order-preserving automorphism of non-orientable

map, if and only if

ξ ∈ Es

ns

E

[1,sn−1

s ].




Assume the cycle index of ξ is [1k 1 , 2k 2 , ..., nk n]. If there exist two integers k i, k j 0

and i, j ≥ 2, i j, then in the cyclic decomposition of ξ , there are two cycles

(u

1, u

2, ..., u

i) and (v

1, v

2, ..., v

j).

Since

[l ξ (u1), l ξ (u2)] = i and [l ξ (v1), l ξ (v2)] = j

and i j, we know that ξ ∗ is not an automorphism of embedding by Theorem 5.3.8.

Whence, the cycle index of ξ must be the form of [1k , sl].

Now if k ≥ 2, let (u), (v) be two cycles of length 1 in the cycle decomposition of ξ .

By Theorem 5.3.8, we know that

[l ξ (u), l ξ (v)] = 1.

If there is a cycle (w,...) in the cyclic decomposition of ξ whose length greater or equal to

2, we get that

[l ξ (u), l ξ (w)] = [1, l ξ (w)] = l ξ (w).

According to Lemma 7.2.1, we get that l ξ (w) = 1, a contradiction. Therefore, the cycle

index of ξ must be the forms of [sl] or [1, sl]. Whence, sl = n or sl + 1 = n. Calculation

shows that l = n

sor l = n−1

s. That is, the cycle index of ξ is one of the following three

types [1n], [1, sn−1

s ] and [sns ] for some integer s ≥ 1.

Now we only need to prove that for each element ξ in E[1,s

n−1s ]

and E[s

ns ]

, there exists

an non-orientable complete map M of order n with the induced permutation ξ ∗ being its

cyclic order-preserving automorphism of surface. The discussion are divided into two

cases.

Case 1. ξ ∈ E[s

ns ]

Assume the cycle decomposition of ξ being ξ = (a, b, · · · , c) · · · ( x, y, · · · , z) · · · (u, v,

· · · , w), where the length of each cycle is k and 1 ≤ a, b, · · · , c, x, y, · · · , z, u, v, · · · , w ≤ n.

In this case, we construct a non-orientable complete map M 1 = (X1α,β

, P1) by defining

X 1α,β = i j+ : 1 ≤ i, j ≤ n, i( j

i j− : 1 ≤ i, j ≤ n, i j,

P1 =

x∈a,b,···,c,···, x, y,···, z,u,v,···,w(C ( x))(αC ( x)−1 α),




where

C ( x) = ( xa+, · · · , x x∗, · · · , xu+, xb+, x y+, · · · , · · · , xv+, xc+, · · · , x z+, · · · , xw+),

x x∗ denotes an empty position and

αC ( x)−1α = ( xa−, xw−, · · · , x z−, · · · , xc−, xv−, · · · , xb−, xu−, · · · , x y−, · · ·).

It is clear that M ξ ∗

1 = M 1. Therefore, ξ ∗ is an cyclic order-preserving automorphism

of map M 1.

Case 2. ξ ∈ E[1,s

n−1s ]

We assume the cyclic decomposition of ξ being that

ξ = (a, b, ..., c)...( x, y, ..., z)...(u, v, ..., w)(t ),

where, the length of each cycle is k beside the final cycle, and 1 ≤ a, b...c, x, y..., z,

u, v, ..., w, t ≤ n. In this case, we construct a non-orientable complete map M 2 = (X 2α,β

, P2)

by defining

X 2α,β = i j+ : 1 ≤ i, j ≤ n, i j

i j− : 1 ≤ i, j ≤ n, i j,

P2 = ( A)(α A−1) x∈a,b,...,c,..., x, y,... z,u,v,...,w(C ( x))(αC ( x)−1α),

where

A = (t a+, t x+, ...t u+, t b+, t y+, ..., t v+, ..., t c+, t z+, ..., t w+),

α A−1α = (t a−, t w−,...t z−, t c−, t v−, ..., t y−, ..., t b−, t u−, ..., t x−),

C ( x) = ( xa+, ..., x x∗, ..., xu+, xb+, ..., x y+, ..., xv+, ..., xc+, ..., x z+, ..., xw+)

and

αC ( x)−1α = ( xa−, xw−,.., x z−, ..., xc−, ..., xv−, ..., x y−, ..., xb−, xu−, ...).

It is also clear that M ξ ∗2 = M 2. Therefore, ξ ∗ is an automorphism of a map M 2 .

Now we consider the case of orientation-reversing automorphisms of complete maps.

According to Lemma 7.2.4, we know that an element ξα, where ξ ∈ S V (K n) is an orientation-

reversing automorphism of complete map only if,

ξ ∈ E[k

n1k ,(2k )

n−n12k ]

.




Our discussion is divided into two parts.

Case 3. n1 = n.

Without loss of generality, we can assume the cycle decomposition of ξ has the

following form in this case.

ξ = (1, 2, · · · , k )(k + 1, k + 2, · · · , 2k ) · · · (n − k + 1, n − k + 2, · · · , n).

Subcase 3.1 k ≡ 1(mod 2) and k > 1.

According to Lemma 7.2.4, we know that ξ ∗α is not an automorphism of map since

o( ξ ∗) = k ≡ 1(mod 2).

Subcase 3.2 k ≡ 0(mod 2).

Construct a non-orientable map M 3 = (X 3α,β

, P3), where X 3 = E (K n) by

P3 =

i∈1,2,···,n(C (i))(αC (i)−1α),

where if i ≡ 1(mod2), then

C (i) = (i1+, ik +1+, · · · , in−k +1+, i2+, · · · , in−k +2+, · · · , ii∗, · · · , ik +, i2k +, · · · , in+),

αC (i)−1α = (i1−, in−, · · · , i2k −, ik −, · · · , ik +1−)

and if i ≡ 0(mod2), then

C (i) = (i1−, ik +1−, · · · , in−k +1−, i2−, · · · , in−k +2−, · · · , ii∗, · · · , ik −, i2k −, · · · , in−),

αC (i)−1α = (i1+, in+, · · · , i2k +, ik +, · · · , ik +1+),

where, ii∗ denotes the empty position, for example, (21, 22∗, 23, 24, 25) = (21, 23, 24, 25). It

is clear that P ξα

3= P3, that is, ξα is an automorphism of map M 3.

Case 4. n1 n.

Without loss of generality, we can assume that

ξ = (1, 2, · · · , k )(k + 1, k + 2, · · · , n1) · · · (n1 − k + 1, n1 − k + 2, · · · , n1)

× (n1 + 1, n1 + 2, · · · , n1 + 2k )(n1 + 2k + 1, · · · , n1 + 4k ) · · · (n − 2k + 1, · · · , n)

Subcase 4.1 k ≡ 0(mod2).




Consider the orbits of 12+ and n1 + 2k + 11+ under the action of ξα, we get that

|orb((12+)<ξα>)| = k

and|orb(((n1 + 2k + 1)1+)<ξα>)| = 2k .

Contradicts to Lemma 7.2.1.

Subcase 4.2 k ≡ 1(mod2).

In this case, if k 1, then k ≥ 3. Similar to the discussion of Subcase 3.1, we know

that ξα is not an automorphism of complete map. Whence, k = 1 and

ξ ∈ E[1n1 ,2n2 ].

Without loss of generality, assume that

ξ = (1)(2) · · · (n1)(n1 + 1, n1 + 2)(n1 + 3, n1 + 4) · · · (n1 + n2 − 1, n1 + n2).

If n2 ≥ 2, and there exists a map M = (X α,β, P), assume a vertex v1 in M being

v1 = (1l12+, 1l13+, · · · , 1l1n+)(1l12−, 1l1n−, · · · , 1l13−)

where, l1i ∈ +2, −2, +3, −3, · · · , +n, −n and l1i l1 j if i j. Then we get that

(v1) ξα = (1l12−, 1l13−, · · · , 1l1n−)(1l12 +, 1l1n+, · · · , 1l13+) v1.

Whence, ξα is not an automorphism of map M , a contradiction. Therefore, n2 = 1.

Similarly, we can also get that n1 = 2. Whence, ξ = (1)(2)(34) and n = 4. We construct a

stable non-orientable map M 4 under the action of ξαby defining

M 4 = (X 4α,β, P4),

where,

P4 = (12+

, 13+

, 14+

)(21+

, 23+

, 24+

)(31+

, 32+

, 34+

)(41+

, 42+

, 43+

)

× (12−, 14−, 13−)(21−, 24−, 23−)(31−, 34−, 32−)(41−, 43−, 42−).

Therefore, all orientation-preserving automorphisms of non-orientable complete maps

are extended actions of elements in

E[s

ns ]

, E[1,s

n−1s ]




and all orientation-reversing automorphisms of non-orientable complete maps are ex-

tended actions of elements in

α

E[(2s)

n2s

]

, α

E[(2s)

4

2s ]

α

E[1,1,2].


According to the Rotation Embedding Scheme for orientable embedding of a graph,

presented by Heff ter firstly in 1891 and formalized by Edmonds in [Edm1], an orientable

complete map is just the case of eliminating the sign + and - in our representation for

complete maps. Whence, we get the following result for automorphism of orientable

complete maps.

Theorem 7.2.2 All orientation-preserving automorphisms of orientable complete maps

of order n ≥ 4 are extended actions of elements in

E[s

ns ]

, E[1,s

n−1s ]

and all orientation-reversing automorphisms of orientable complete maps of order n ≥ 4

are extended actions of elements in

αE[(2s)

n2s ]

, αE[(2s)

42s ]

, αE[1,1,2],

where,E

θ denotes the conjugatcy class containing θ in S V (K n).

Proof The proof is similar to that of Theorem 7.2.1. For completion, we only need

to construct orientable maps M Oi

, i = 1, 2, 3, 4 to replace non-orientable maps M i, i =

1, 2, 3, 4 in the proof of Theorem 7.2.1. In fact, for orientation-preserving cases, we only

need to take M O1

, M O2

to be the resultant maps eliminating the sign + and - in M 1, M 2

constructed in the proof of Theorem 7.2.1. For the orientation-reversing cases, we take

M O3

= ( E (K n)α,β, P O3

) with

P3 =

i

∈1,2,

···,n

(C (i)),

where, if i ≡ 1(mod2), then

C (i) = (i1, ik +1, · · · , in−k +1, i2, · · · , in−k +2, · · · , ii∗, · · · , ik , i2k , · · · , in),

and if i ≡ 0(mod2), then

C (i) = (i1, ik +1, · · · , in−k +1, i2, · · · , in−k +2, · · · , ii∗, · · · , ik , i2k , · · · , in)−1,



Sec.7.3 Map-Automorphism Graphs 257

where ii∗ denotes the empty position and M O4

= ( E (K 4)α,β, P4) with

P4 = (12, 13, 14)(21, 23, 24)(31, 34, 32)(41, 42, 43).

It can be shown that ( M Oi ) ξ ∗α = M Oi for i = 1, 2, 3 and 4.

§7.3 MAP-AUTOMORPHISM GRAPHS

7.3.1 Semi-Regular Graph. A graph is called to be a semi-regular graph if it is simple

and its automorphism group action on its ordered pair of adjacent vertices is fixed-free,

which is considered in [Mao1] and [MLT1] for enumerating its non-equivalent embed-

dings on surfaces. A map underlying a semi-regular graph is called to be a semi-regular map. We determine all automorphisms of maps underlying a semi-regular graph in this

section.

Comparing with the Theorem 7.1.2, we get a necessary and sufficient condition for

an automorphism of a graph being that of a map.

Theorem 7.3.1 For a connected graph G, an automorphism ξ ∈ AutG is an orientation-

preserving automorphism of non-orientable map underlying graph G if and only if ξ is

semi-regular acting on its ordered pairs of adjacent vertices.

Proof According to Theorem 5.3.5, if ξ ∈ AutG is an orientation-preserving auto-

morphism of map M underlying graph G, then ξ is semi-regular acting on its ordered pairs

of adjacent vertices.

Now assume that ξ ∈ AutG is semi-regular action on its ordered pairs of adjacent

vertices. Denote by ξ |V (G), ξ | E (G) β the action of ξ on V (G) and on its ordered pairs of

adjacent vertices, respectively. By conditions in this theorem, we can assume that

ξ |V (G) = (a, b, · · · , c) · · · (g, h, · · · , k ) · · · ( x, y, · · · , z)

and

ξ | E (G) β = C 1 · · · C i · · · C m,

where, let sa = |a, b, · · · , c|, · · ·, sg = |g, h, · · · , k |, · · ·, s x = | x, y, · · · , z|, then sa|C (a)| =

· · · = sg|C (g)| = · · · = s x|C ( x)|, and C(g) denotes the cycle containing g in ξ |V (G) and

C 1 = (a1, b1, · · · , c1, a2, b2, · · · , c2, · · · , asa, bsa , · · · , csa ),




· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·,C i = (g1, h1, · · · , k 1, g2, k 2, · · · , k 2, · · · , gsg , hsg, · · · , k sg ),

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·,

C m = ( x1, y1, · · · , z1, · · · , x2, y2, · · · , z2, · · · , xs x , ys x , · · · , zs x ).

Now for ∀ ξ,ξ ∈ AutG, we construct a stable map M = (X α,β, P) under the action

of ξ as follows.

X = E (Γ)

and

P =g∈T V

ξ

x∈C (g)

(C x)(αC −1 x ).

Assume that u = ξ f (g), and

N G(g) = g z1, g z2 , · · · , g zl.

Obviously, all degrees of vertices in C (g) are same. Notices that ξ | N G(g) is circular acting

on N G(g) by Theorem 7.1.2. Whence, it is semi-regular acting on N G(g). Without loss of

generality, we assume that

ξ | N G(g) = (g z1 , g z2 , · · · , g zs)(g zs+1 , g zs+2 , · · · , g z2s ) · · · (g z(k −1)s+1, g z(k −1)s+2 , · · · , g zks ),

where, l = ks. Choose

C g = (g z1+, g zs+1+, · · · , g z(k −1)s+1+, g z2+, g zs+2+, · · · , g zs +, g z2s, · · · , g zks +).

Then,

C x = ( x z1 +, x zs+1+, · · · , x z(k −1)s+1+, x z2+, x zs+2 +, · · · , x zs+, x z2s , · · · , x zks +),

where,

x zi+= ξ f (g zi+),

for i = 1, 2, · · · , ks. and

αC −1 x = (α x z1+, α x zs+1+, · · · , α x z(k −1)s+1+, α x zs+, α x z2s , · · · , α x zks +).

Immediately, we get that M ξ = ξ M ξ −1 = M by this construction. Whence, ξ is an

orientation-preserving automorphism of map M .




By the rotation embedding scheme, eliminating α on each quadricell in Tutte’s rep-

resentation of embeddings induces an orientable embedding underlying the same graph.

Since an automorphism of embedding is commutative with α and β, we get the follow-

ing result for the orientable-preserving automorphisms of orientable maps underlying asemi-regular graph.

Theorem 7.3.2 If G is a connected semi-regular graph, then for ∀ ξ ∈ AutG, ξ is an

orientation-preserving automorphism of orientable map underlying graph G.

According to Theorems 7.3.1 and 7.3.2, if G is semi-regular, i.e., each automor-

phism acting on the ordered pairs of adjacent vertices in G is fixed-free, then every auto-

morphism of graph G is an orientation-preserving automorphism of orientable map and

non-orientable map underlying graph G. We restated this result in the following.

Theorem 7.3.3 If G is a connected semi-regular graph, then for ∀ ξ ∈ AutG, ξ is an

orientation-preserving automorphism of orientable map and non-orientable map under-

lying graph G.

Notice that if ς ∗ is an orientation-reversing automorphism of map, then ς ∗α is an

orientation-preserving automorphism of the same map. By Lemma 7.2.4, if τ is an auto-

morphism of map underlying a graph G, then τα is an automorphism of map underlying

this graph if and only if o(τ) ≡ 0(mod2). Whence, we have the following result for

automorphisms of maps underlying a semi-regular graph

Theorem 7.3.4 Let G be a semi-regular graph. Then all the automorphisms of orientable

maps underlying graph Γ are

g|X α,β and αh|X α,β , g, h ∈ AutG with o(h) ≡ 0(mod2).

and all the automorphisms of non-orientable maps underlying graph G are also

g

|X α,β and αh

|X α,β , g, h

∈AutΓ with o(h)

≡0(mod2).

Theorem 7.3.4 will be used in Chapter 8 for the enumeration of maps on surfaces

underlying a semi-regular graph.

An circulant transitive graph of prime order is Cayley graph Cay( Z p : S ), B.Alspach

completely determined its automorphism group as follows([Als1]):




If S = ∅ , or S = Z ∗ p , then Aut(Cay( Z p : S )) =

p , the symmetric group of degree p,

otherwise,

Aut(Cay( Z p : S )) = T a,b|a ∈ H , b ∈ Z ∗ p,

where T a,b is the permutation on Z p which maps x to ax +b and H is the largest even order

subgroup of Z ∗ p such that S is a union of cosets of H.

We get a corollary from Theorem 7.3.4 for circulants of prime order.

Corollary 7.3.1 Every automorphism of a circulant graph G, not be a complete graph,

with prime order is an orientation-preserving automorphism of map underlying graph G

on orientable surfaces.

Proof According to Theorem 7.3.4, we only need proving that each automorphism

θ = ax + b of the circulant graph Cay( Z p : S ), Cay( Z p : S ) K n is semi-regular acting

on its order pairs of adjacent vertices, where p is a prime number. Now for an arc gsg =

(g, sg) ∈ A(Cay( Z p : S )), where A(G) denotes the arc set of the graph Γ, we have that

(gsg)θ = (ag + b)asg+b;

(gsg)θ 2 = (a(ag + b) + b)a(asg+b)+b = (a2g + ab + b)a2 sg+ab+b;

· · · · · · · · · · · · · · · · · · · · · · · · ;

(gsg)θ o(a)

= (ao(a)g + ao(a)−1b + ao(a)−2b + · · · + b)ao(a) sg+ao(a)−1b+ao(a)−2b+···+b

= (a

o(a)

g +

ao(a)b

−1

a − 1 )ao(a) sg+ ao(a)b−1

a

−1

= g

sg

,

where o(a) denotes the order of a. Therefore, θ is semi-regular acting on the order pairs

of adjacent vertices of the graph Cay( Z p : S ).

For symmetric circulant of prime order, not being a complete graph, Chao proved

that the automorphism group is regular acting on its order pairs of adjacent vertices([Cha1]).

Whence, we get the following result.

Corollary 7.3.2 Every automorphism of a symmetric circulant graph G of prime order

p, G K p , is an orientation-preserving automorphism of map on orientable surface

underlying graph G.

Now let s be an even divisor of q − 1 and r a divisor of p − 1. Choose H ( p, r ) =< a >

, t ∈ Z ∗ p be such that t s2 ∈ − H ( p, r ) and u the least common multiple of r and the order of

t in Z ∗ p. The graph G( pq; r , s, u) is defined as follows:

V (G( pq; r , s, u)) = Z q × Z p = (i, x)|i ∈ Z q, x ∈ Z p.




E (G( pq; r , s, u)) = ((i. x), ( j, y))|∃l ∈ Z +such that j − i = al, y − x ∈ t l H ( p, r ).

It is proved that the automorphism group of G( pq; r , s, u) is regular acting on the

ordered pairs of adjacent pairs in [PWX1]. By Theorem 7.3.4, we get the following

result.

Corollary 7.3.3 Every automorphism of graph G( pq; r , s, u) is an orientation-preserving

automorphism of map on orientable surface underlying graph G( pq; r , s, u).

7.3.2 Map-Automorphism Graph. A graph G is a map-automorphism graph if all

automorphisms of G is that of maps underlying graph G. Whence, every semi-regular

graph is a map-automorphism graph. According to Theorems 7.1.1-7.1.2, we know the

following result.

Theorem 7.3.5 A graph G is a map-automorphism graph if and only if for ∀v ∈ V (G) ,

the stabilizer (AutG)v ≤ v × α.

Proof By definition, G is a map-automorphism graph if all automorphisms of G

are automorphisms of maps underlying G, i.e., AutG is an automorphism group of map.

According to Theorems 7.1.1 and 7.1.2, we know that this happens if and only if for

∀v ∈ V (G), the stabilizer (AutG)v ≤ v × α.

We therefore get the following result again.

Theorem 7.3.6 Every semi-regular graph G is a map-automorphism graph.

Proof In fact, we know that (AutG)v = 1V (G) ≤ v × α for a semi-regular graph G.

By Theorem 7.3.5, G is a map-automorphism graph.

Further application of Theorem 7.3.6 enables us to get the following result for vertex

transitive graphs.

Theorem 7.3.7 A Cayley graph X = Cay(Γ : S ) is a map-automorphism graph if and

only if (Aut X )1Γ ≤ (S ) , where (S ) denotes a cyclic permutation on S . Furthermore, thereis a regular map underlying Cay(Γ : S ) if (Aut X )1Γ

≤ (S ).

Proof Notice that a Cayley graph Cay(Γ : S ) is transitive by Theorem 3.2.1. For

∀g, h ∈ V (Cay(Γ : S )), such a transitive automorphism is τ = g−1 h : g → h. We

therefore know that (Aut X )g ≃ (Aut X )h for g, h ∈ V (Cay(Γ : S )). Whence, X is a map-

automorphism graph if and only if (Aut X )1Γ≤ (S ) by Theorem 7.3.6. In this case, there is




a regular map underlying Cay(Γ : S ) was verified by Gardiner et al. in [GNSS1], seeing

Note 7.1.1.

Particularly, we get the following conclusion for map-automorphism graphs.

Corollary 7.3.4 A GRR graph of a finite group (Γ; ) is a map-automorphism graph.

Corollary 7.3.5 A Cayley map Cay M (Γ : S , r ) is regular if and only if there is an auto-

morphism τ ∈ AutΓ such that τ|S = r.

Proof This is an immediately conclusion of Theorems 5.4.7 and 7.3.7.

A few map-automorphism graphs can be found in Table 7.3.1 following.

G AutG Map-automorphism Graph?

Pn Z 2 Yes

C n Dn Yes

Pn × P2 Z 2 × Z 2 Yes

C n × P2 Dn × Z 2 Yes

Table 7.3.1

§7.4 AUTOMORPHISMS OF ONE FACE MAPS

7.4.1 One-Face Map. A one face map is such a map just with one face, which means

that the underlying graph of one face maps is the bouquets. Therefore, for determining

the automorphisms of one face maps, we only need to determine the automorphisms of

bouquets Bn on surfaces. There is a well-know result for automorphisms of a map and its

dual in topological graph theory, i.e., the automorphism group of map is the same as its

dual.

A map underlying graph Bn for an integer n ≥ 1 has the form Bn = (X α,β, Pn) with

X = E ( Bn) = e1, e2, · · · , en and

Pn = ( x1, x2, · · · , x2n)(α x1, α x2n, · · · , x2),

where, xi ∈ X , β X or αβ X and satisfying Axioms 1 and 2 in Section 5.2 of Chapter 5. For

a given bouquet Bn with n edges, its semi-arc automorphism group is

Aut 12 Bn = S n[S 2].



Sec.7.4 Automorphisms of One Face Maps 263

From group theory, we know that each element in S n[S 2] can be represented by (g; h1, h2,

· · · , hn) with g ∈ S n and hi ∈ S 2 = 1, αβ for i = 1, 2, · · · , n. The action of (g; h1, h2, · · · , hn)

on a map Bn underlying graph Bn by the following rule:

If x ∈ ei, αei, βei, αβei , then (g; h1, h2, · · · , hn)( x) = g(hi( x)).

For example, if h1 = αβ, then, (g; h1, h2, · · · , hn)(e1) = αβg(e1), (g; h1, h2, · · · , hn)(αe1)

= βg(e1), (g; h1, h2, · · · , hn)( βe1) = αg(e1) and (g; h1, h2, · · · , hn)(αβe1) = g(e1).

The following result for automorphisms of a map underlying graph Bn is obvious.

Lemma 7.4.1 Let (g; h1, h2, · · · , hn) be an automorphism of map Bn underlying a graph

Bn. Then

(g; h1, h2, · · · , hn) = ( x1, x2, ..., x2n)k

and if (g; h1, h2, · · · , hn)α is an automorphism of map Bn , then

(g; h1, h2, · · · , hn)α = ( x1, x2, · · · , x2n)k

for some integer k , 1 ≤ k ≤ n, where xi ∈ e1, e2, · · · , en , i = 1, 2, · · · , 2n and xi x j if

i j.

7.4.2 Automorphisms of One-Face Map. Analyzing the structure of elements in group

S n[S 2], we get the automorphisms of maps underlying graph Bn by Theorems 7.3.1 and

7.3.2 as follows.

Theorem 7.4.1 Let Bn be a bouquet with n edges ei for i = 1, 2, · · · , n. Then the automor-

phisms (g; h1, h2, · · · , hn) of orientable maps underlying Bn for n ≥ 1 are respectively

(O1) g ∈ E[k

nk ]

, hi = 1, i = 1, 2, · · · , n;

(O2) g ∈ E[k

nk ]

and if g =

n/k i=1

(i1, i2, · · · ik ), where i j ∈ 1, 2, · · · , n, n/k ≡ 0(mod2),

then hi1= (1, αβ), i = 1, 2, · · · , n

k and hi j

= 1 f o r j ≥ 2;

(O3) g ∈ E[k 2s ,(2k )

n−2ks2k ]

and if g =

2s

i=1

(i1, i2, · · · ik )

(n−2ks)/2k

j=1

(e j1, e j2

, · · · , e j2k ), where

i j, e jt ∈ 1, 2, · · · , n, then hi1

= (1, αβ), i = 1, 2, · · · , s, hil= 1 for l ≥ 2 and h jt

=

1 f or t = 1, 2, · · · , 2k ,

and the automorphisms (g; h1, h2, · · · , hn) of non-orientable maps underlying Bn for n ≥ 1

are respectively

( N 1) g ∈ E[k

nk ]

, hi = 1, i = 1, 2, · · · , n;




( N 2) g ∈ E[k

nk ]

and if g =

n/k i=1

(i1, i2, · · · ik ), where i j ∈ 1, 2, · · · , n, n/k ≡ 0(mod2),

then hi1= (1, αβ), (1, β) with at least one hi0 1

= (1, β) for i = 1, 2, · · · , nk

and hi j= 1 for j ≥

2;

( N 3) g ∈ E[k 2s ,(2k )

n−2ks2k ]

and if g =

2si=1

(i1, i2, · · · ik )

(n−2ks)/2k j=1

(e j1, e j2

, · · · , e j2k ), where

i j, e jt ∈ 1, 2, · · · , n, then hi1

= (1, αβ), (1, β) with at least one hi0 1= (1, β) f o r i =

1, 2, · · · , s and hil= 1 f o r l ≥ 2 and h jt

= 1, t = 1, 2, · · · , 2k , where Eθ denotes the

conjugacy class in symmetry group S V (Bn) containing the element θ .

Proof By the structure of group S n[S 2], it is clear that the elements in the cases

(1), (2) and (3) are all semi-regular. We only need to construct an orientable or non-

orientable mapB

n = (X α,β, Pn) underlying Bn stable under the action of elements in

each case.

(1) g =

n/k i=1

(i1, i2, · · · ik ) and hi = 1, i = 1, 2, · · · , n, where i j ∈ 1, 2, · · · , n.

Choose

X 1α,β =

n/k i=1

K i1, i2, · · · , ik ,

where K = 1,α,β,αβ and

P1n = C 1(αC −1

1 α−1)

with

C 1 = ( 11, 21, · · · , (n

k )1, αβ11, αβ21, · · · , αβ(

n

k )1, 12, 22, · · · , (

n

k )2,

αβ12, αβ22, · · · , αβ(n

k )2, · · · , 1k , 2k , · · · , (

n

k )k , αβ1k , αβ1k , · · · , αβ(

n

k )k ).

Then the map B1n = (X 1

α,β, P 1

n ) is an orientable map underlying graph Bn and stable under

the action of (g; h1, h2, · · · , hn).

For the non-orientable case, we chose

C 1 =

11, 21, · · · , (

n

k )1, β11, β21, · · · , β(

n

k )1, 12, 22, · · · , (

n

k )2,

β12, β22, · · · , β(n

k )2, · · · , 1k , 2k , · · · , (

n

k )k , β1k , β1k , · · · , β(

n

k )k

.


α,β, P 1n ) is a non-orientable map underlying graph Bn and stable

under the action of (g; h1, h2, · · · , hn).



Sec.7.4 Automorphisms of One Face Maps 265

(2) g =

n/k i=1

(i1, i2, · · · ik ), hi = (1, β) or (1, αβ), i = 1, 2, · · · , n, nk

≡ 0(mod2), where

i j ∈ 1, 2, · · · , n.

If hi1 = (1, αβ) for i = 1, 2, · · · ,n

k and hit = 1 for t ≥ 2, then

(g; h1, h2, · · · , hn) =

n/k i=1

(i1, αβi2, · · · αβik , αβi1, i2, · · · , ik ).

Similar to the case of (1), let X 2α,β = X 1

α,β and

P 2n = C 2(αC −1

2 α−1)

with

C 2 =

11, 21, · · · , (

n

k )1, αβ12, αβ22, · · · , αβ(

n

k )2, αβ1k , αβ2k ,

· · · , αβ( nk

)k , αβ11, αβ21, · · · , αβ( nk

)1, 12, 22, · · · , ( nk

)2, · · · , 1k , 2k , · · · , ( nk

)k .


α,β, P2n ) is an orientable map underlying graph Bn and stable under

the action of (g; h1, h2, · · · , hn). For the non-orientable case, the construction is similar.

Now it only need to replace each element αβi j by that of βi j in the construction of the

orientable case if hi j= (1, β).

(3) g =

2si=1

(i1, i2, · · · ik )

(n−2ks)/2k j=1

(e j1, e j2

, · · · , e j2k ) and hi1

= (1, αβ), i = 1, 2, · · · , s,

hil = 1 for l ≥ 2 and h jt = 1 for t = 1, 2, · · · , 2k .In this case, we know that

(g; h1, h2, · · · , hn) =

si=1

(i1, αβi2, · · · αβik , αβi1, i2, · · · , ik )

(n−2ks)/2k j=1

(e j1, e j2

, · · · , e j2k ).

Denote by p the number (n − 2ks)/2k . We construct an orientable map B3n = (X 3

α,β, P 3

n )

underlying Bn stable under the action of (g; h1, h2, · · · , hn) as follows.

Take

X 3α,β = X 1

α,β and P 3n = C 3(αC −1

3 α−1)

with

C 3 =11, 21, · · · , s1, e11

, e21, · · · , e p1

, αβ12, αβ22, · · · , αβs2,

e12, e22

, · · · , e p2, · · · , αβ1k , αβ2k , · · · , αβsk , e1k

, e2k , · · · ,

e pk , αβ11, αβ21, · · · , αβs1, e1k +1

, e2k +1, · · · , e pk +1

, 12, 22, · · · ,

s2, e1k +2, e2k +2

, · · · , e pk +2, · · · , 1k , 2k , · · · , sk , e12k

, e22k , · · · , e p2k

.





α,β, P 3

n ) is an orientable map underlying graph Bn and stable under

the action of (g; h1, h2, · · · , hn).

Similarly, replacing each element αβi j by βi j in the construction of the orientable

case if hi j = (1, β), a non-orientable map underlying graph Bn and stable under the actionof (g; h1, h2, · · · , hn) can be also constructed. This completes the proof.

We will apply Theorem 7.4.1 for the enumeration of one face maps on surfaces in

Chapter 8.

§7.5 REMARKS

7.5.1 An automorphism of map M is an automorphism of graph underlying that of M .But the conversely is not always true. Any map automorphism is fixed-free, i.e., semi-

regular, particularly, an automorphism of regular map is regular. This fact enables one

to characterize those automorphisms of maps underlying a graph. Certainly, there is an

naturally induced action g|X α,β for an automorphism g ∈ AutG of graph G on quadricells

in maps underlying G, i.e.,

(α x)g = α y, ( β x)g = β y, (αβ x)g = αβ y

if xg = y for ∀ x ∈ X α,β( M (G)). Consider the action of AutG on X α,β( M (G)). Then we

get the following result by definition.

Theorem 7.5.1 An automorphism g of G is a map automorphism if and only if there is a

map M (G) stabilized under the action of g|X α,β .

Theorems 7.1.1 and 7.1.2 enables one to characterize such map automorphisms in

another way, i.e., the following.

Theorem 7.5.2 An automorphism g∈

AutG of graph G is an automorphism of map

underlying G if and only if gv ≤ v × α for ∀v ∈ V (G).

7.5.2 We get these permutation presentations for automorphisms of maps underlying a

complete graph, a semi-regular graph and a bouquet, which enables us to calculate the

stabilizer Φ(g) of g on maps underlying such a graph in Chapter 8. A general problem is

the following.



Sec.7.5 Remarks 267

Problem 7.5.1 Find a permutation presentation for map automorphisms induced by such

automorphisms of a graph G on quadricells X α,β with base set X = E (G) , particularly,

find such presentations for complete bipartite graphs, cubes, generalized Petersen graphs

or regular graphs in general.

7.5.3 We had introduced graph multigroup for characterizing the local symmetry of a

graph, i.e., let G be a connected graph, H ≤ G a connected subgraph and τ ∈ AutG.

Similarly, consider the induced action of τ on X α,β with base set X = E ( H ). Then the

following problem is needed to answer.

Problem 7.5.2 Characterize automorphisms of maps underlying H induced by auto-

morphisms of graph G, or verse via, characterize automorphisms of maps underlying G

induced by automorphisms of graph H by introducing the action of Aut H on G \ H witha stabilizer H.



CHAPTER 8.

Enumerating Maps on Surfaces

There are two kind of maps usually considered for enumeration in literature.One is the rooted map, i.e., a quadricell on map marked beforehand. Such a

map is symmetry-freed, i.e., its automorphism group is trivial. Another is the

map without roots marked. The enumeration of maps on surfaces underlying

a graph can be carried out by the following programming:

STEP 1. Determine all automorphisms g of maps underlying graph G;

STEP 2. Calculate the the fixing set Φ1(g) or Ψ2(g) for each automorphism

g ∈ Aut 12

G;

STEP 3. Enumerate the maps on surfaces underlying graph G by Burnsidelemma.

This approach is independent on the orientability of maps. So it enables one to

enumerate orientable or non-orientable maps on surfaces both. The roots dis-

tribution and a formula for rooted maps underlying a graph are included in the

first two sections. Then a general enumeration scheme for maps underlying a

graph is introduced in Section 3. By applying this scheme, the enumeration

formulae for maps underlying a complete graph, a semi-regular graph or a

bouquet are obtained by applying automorphisms of maps determined in lastchapter in Sections 8.3-8.6, respectively.



Sec.8.1 Roots Distribution on Embeddings 269

§8.1 ROOTS DISTRIBUTION ON EMBEDDINGS

8.1.1 Roots on Embedding. A root of am embedding M = (X α,β, P) of graph G is an

element in X α,β. A root r is called an i-root if it is incident with a vertex of valency i. Two

i-roots r 1, r 2 are transitive if there exists τ ∈ Aut M such that τ(r 1) = r 2. An enumerator

v( D, x) and the root polynomials r ( M , x), r (M( D), x) of M are defined by

v( D, x) =i≥1

ivi xi;

r ( M , x) =i≥1

r ( M , i) xi,

where r ( M , i) denotes the number of non-transitive i-roots in M and

r (M( D), x) = M ∈M( D)

r ( M , x).

Theorem 8.1.1 For any embedding M (orientable or non-orientable),

r ( M , i) =2ivi

|AutM| ,

where vi denotes the number of vertices with valency i in M.

Proof Let U be all i-roots on M . Since U AutM = U , AutM is also a permutation

group acting on U , and r ( M , i) is the number of orbits in U under the action of AutM.

It is clear that |U | = 2ivi. For ∀r ∈ U , (AutM)r is the trivial group by Theorem 5.3.5.According to Theorem 2.1.1(3), |AutM| = |(AutM)r ||r AutM |, we get that |r AutM | = |AutM|.Thus the length of each orbit inU under this action has |AutM| elements. Whence,

r ( M , i) =|U |

|AutM| =2ivi

|AutM| .

Applying Theorem 8.1.1, we get a relation between v( D, x) and r ( M , x) following.

Theorem 8.1.2 For an embedding M (orientable or non-orientable ) with valency se-

quence D,

r ( M , x) = 2v( D, x)|AutM| .

Proof By Theorem 8.1.1, we know that r ( M , i) =2ivi

|AutM| , where vi denotes the

number of vertices of valency i in M . So we have

r ( M , x) =i≥1

r ( M , i) xi



270 Chap.8 Enumerating Maps on Surfaces

=i≥1

2ivi

|AutM| =2v( D, x)

|AutM|

Let r ( M ) denotes the number of non-transitive roots on an embedding M . As a by-

product, we get r ( M ) by Theorem 8.1.2 following.

Corollary 8.1.1 For a given embedding M,

r ( M ) =4ε( M )

|AutM| ,

where ε( M ) denotes the number of edges of M.

Proof According to Theorem 8.1.2, we know that

r ( M ) = r ( M , 1) =2v( D, 1)

|AutM|=

1

|AutM|i≥1

2ivi.

Noticei≥1

ivi = 2ε( M ). We get that

r ( M ) =4ε( M )

|AutM| .

8.1.2 Root Distribution. Let G be a connected simple graph and D = d 1, d 2, · · · , d vits valency sequence. For ∀g ∈ AutG, there is an extended action g|X α,β acting on X α,β

with X = E (G). Define the orientable embedding index θ O(G) of G and the orientable

embedding index θ O

( D) of D respectively by

θ O(G) =

M ∈M(G)

1

|AutM| ,

θ O( D) =

G∈G( D)

M ∈M(G)

1

|AutM| ,

where G( D) denotes the family of graphs with valency sequence D. Then we have the

following results.

Theorem 8.1.3 For any connected simple graph G and a valency sequence D ,

θ O(G) =

d ∈ D(G)

(d − 1)!

2|AutG| and θ O( D) =

d ∈ D(G)

(d − 1)!

2|∆( D)| ,

where

|∆( D)|−1 =

G∈G( D)

1

|AutG| .




Proof Let W be the set of all embedings of graph G on orientable surfaces. Since

there is a bijection between the rotation scheme set (G) of G and W , it is clear that

|W | = |

(G)| =

d

∈ D(G)

(d − 1)!. Notice that every element ξ ∈ AutG naturally induces an

g|X α,β action on W . Since for an embeding M , ξ ∈ AutM if and only if ξ ∈ (AutG × α) M ,

so AutM = (AutG × α) M . By |AutG × α | = |(AutG × α) M || M AutG×α|, we get that

| M AutG×α| =|AutG × α |

|Aut M | .

Therefore, we have that

θ O(G) =

M ∈M(G)

1

|AutM|

= 1|AutG × α | M ∈M(G)

|AutG × α ||AutM|

=1

|AutG|

M ∈M(G)

| M AutG×α|

=|W |

2|AutG| =

d ∈ D(G)

(d − 1)!

2|AutG|and

θ O( D) = G∈G( D)

d ∈ D(G)

(d

−1)!

2|AutG|)

=1

2

d ∈ D(G)

(d − 1)!(

G∈G( D)

1

|AutG|)

=

d ∈ D(G)

(d − 1)!

2|∆( D)| .

Now we prove the main result of this subsection.

Theorem 8.1.4 For a given valency sequence D =

d 1, d 2,

· · ·, d v

,

r (M( D), x) =

v( D, x)

d ∈ D(G)(d − 1)!

|∆( D)| .

where,

|∆( D)|−1 =

G∈G(D)

1

|AutG| .




Proof By the definition of r (M( D), x), we know that

r (M( D), x) =

M ∈M( D)

r ( M , x)

= G∈G( D)

M ∈M(G)

r ( M , x).

According to Theorem 8.1.3, we know that

r (M( D), x) =

G∈G( D)

M ∈M(G)

2v( D, x)

|AutM| = 2v( D, x)θ ( D).

Whence,

θ ( D) =

d ∈ D(G)

(d − 1)!

2

|∆( D)

|

.

Therefore, we finally get that

r (M( D), x) =

v( D, x)

d ∈ D(G)(d − 1)!

|∆( D)| .

Corollary 8.1.2 For a connected simple graph G, let D(G) = d 1, d 2, · · · , d v be its valency

sequence. Then

r (M(G), x) =

v( D, x)

d ∈ D(G)

(d − 1)!

|AutG| .

Corollary 8.1.4 The number of all non-transitive i-roots in embeddings underlying aconnected simple graph G is

ivi

d ∈ D(G)

(d − 1)!

|AutG| ,

where vi denotes the number of vertices of valency i in G.

Corollary 8.1.5 The number r (M(G)) of non-transitive roots in embeddings of simple

graph G on orientable surfaces is

r (M(G)) =

2ε(G)

d

∈ D(G)

(d − 1)!

|AutG| .

Proof According to Theorem 8.1.2 and Corollary 8.1.2, we know that

r (M(G)) = r (M(G), 1)

=

d ∈ D(G)

(d − 1)!v( D, 1)

|AutG| .




Notice that v( D, 1) =i≥1

ivi = 2ε( M ). So we find that

r (M(G)) =

2ε(G)

d ∈ D(G)(d − 1)!

|AutG| .

Theorem 8.1.4 enables one to enumerate roots on edmeddings underlying a vertex-transitive

graphs, a symmetric graph, · · ·, etc. For example, we can apply Corollary 8.1.5 to count

the roots on embeddings underlying a complete graph K n. In this case, AutKn = S V (K n),

so |AutKn| = n!. Therefore,

r (M(K n)) =n(n − 1)((n − 2)!)n

n!= ((n − 2)!)n−1.

let n = 4. Calculation shows that there are eight non-transitive roots on embeddingsunderlying K 4, shown in the Fig.8.1.1, in which each arrow represents a root.

¹

¹

1

1

2 2

¹

Á

Ê

·

¿

¹

¹

1

1

2 2

Fig.8.1.1

8.1.3 Rooted Map. A rooted map M r is such a map M = (X , P) with one quadricell

r ∈ X α,β is marked beforehand, which is introduced by Tutte for the enumeration of planar

maps. Two rooted maps M r 11 and M r 2

2 are said to be isomorphic if there is an isomorphism

θ : M 1 → M 2 between M 1 and < M 2 such that θ (r 1) = r 2, particularly, if M 1 = M 2 = M ,

two rooted maps M r 1 and M r 2 are isomorphic if and only if there is an automorphism

τ

∈Aut M such that τ(r 1) = r 2. All automorphisms of a rooted map M r form a group,

denoted by Aut M r . By Theorem 5.3.5, we know the following result.

Theorem 8.1.5 Aut M r is a trivial group.

The importance of the idea introduced a root on map is that it turns any map to a

non-symmetry map. The following result enables one to enumerate rooted maps by that

of roots on maps.




Theorem 8.1.6 For a map M = (X α,β, P) , the number of non-isomorphic rooted maps

is equal to that of non-transitive roots on map M.

Proof Let r 1 and r 2 be two non-transitive roots on M . Then M r 1 and M r 2 are non-

isomorphic by definition. Conversely, if M r 1 and M r 2 are non-isomorphic, there are no

automorphisms τ ∈ Aut M such that τ(r 1) = r 2, i.e., r 1 and r 2 are non-transitive. .

Theorem 8.1.6 turns the enumeration of rooted maps by that of roots on maps.

Theorem 8.1.7 The number r O(G) of rooted maps on orientable surfaces underlying a

connected graph G is

r O(G) =

2ε(G)

v∈V (G)( ρ(v) − 1)!

|Aut 12

G| ,

where ρ(v) denotes the valency of vertex v.

Proof Denotes the set of all non-isomorphic orientable maps with underlying graph

G by MO(G). According to Corollary 8.1.1 and Theorem 8.1.6, we know that

r O(G) =

M ∈MO(G)

4ε( M )

|Aut M | .

Notice that every element ξ ∈ AutG 12

× α natural induces an action on EO(G). By

Theorem 5.3.3, ∀ M ∈ M(G), τ ∈ Aut M if and only if, τ ∈ (AutG 12

× α) M . Whence,

Aut M = (AutG 12 × α) M . According to Theorem 2.1.1(3), |AutG 1

2 × α | = |(AutG 12 ×

α) M || M AutG 1

2×α|. We therefore get that

| M AutG 1

2×α| =

2|AutG||Aut M | .

Whence,

r O(G) = 4ε(G)

M ∈MO(G)

1

|Aut M |

= 4ε(G)|AutG 12× α |

M ∈MO(G)

|AutG 1

2 × α ||Aut M |

=4ε(G)

|AutG 12× α |

M ∈MO(G)

| M AutG 1

2×α|

=4ε(G)|EO(G)|

2|AutG 12| =

2ε(G)

v∈V (G)( ρ(v) − 1)!

|Aut 12

G|




By Theorems 3.4.1 and 8.1.7, we get a corollary for the number of rooted orientable

maps underlying a simple graph, which is the same as Corollary 8.1.5 following.

Corollary 8.1.6 The number r O(G) of rooted maps on orientable surfaces underlying a

connected simple graph G is

r O(G) =

2ε( H )

v∈V (G)( ρ(v) − 1)!

|AutG| .

For rooted maps on locally orientable surfaces underlying a connected graph G, we

know the following result.

Theorem 8.1.8 The number r L(G) of rooted maps on surfaces underlying a connected

graph G is

r L(G) =2

β(G)+1

ε(G) v∈V (G)( ρ(v) − 1)!

| Aut 12

G| .

Proof The proof is similar to that of Theorem 8.1.7. In fact, by Corollaries 5.1.2,

8.1.1 and Theorem 8.1.6, let M L(G) be the set of all non-isomorphic maps underlying

graph G. Then

r L(G) =

M ∈M L(G)

4ε( M )

|Aut M | = 4ε(G)

M ∈M L(G)

1

|Aut M |

=

4ε(G)

|AutG 12× α |

M ∈M L(G)|AutG 1

2

× α

||Aut M |

=4ε(G)

|AutG 12× α |

M ∈M L(G)

| M AutG 1

2×α|

=4ε(G)|E L(G)|

2|AutG 12| =

2 β(G)+1ε(G)

v∈V (G)( ρ(v) − 1)!

| Aut 12

G| .


Since r L(G) = r O(G) + r N (G), we also get the number r N (G) of rooted maps on

non-orientable surfaces underlying a connected graph G following.

Theorem 8.1.9 The number r N (G) of rooted maps on non-orientable surfaces underlying

a connected graph G is

r N (G) =

(2 β(G)+1 − 2)ε(G)

v∈V (G)( ρ(v) − 1)!

| Aut 12

G| .




According to Theorems 8.1.8 and 8.1.9, we get the following table for the numbers

of rooted maps on surfaces underlying a few well-known graphs.

G r O(G) r N (G)

Pn n − 1 0

C n 1 1

K n (n − 2)!n−1 (2(n−1)(n−2)

2 − 1)(n − 2)!n−1

K m,n(m n) 2(m − 1)!n−1(n − 1)!m−1 (2mn−m−n+2 − 2)(m − 1)!n−1(n − 1)!m−1

K n,n (n − 1)!2n−2 (2n2−2n+2 − 1)(n − 1)!2n−2

Bn(2n)!

2nn!(2n+1 − 1) (2n)!

2nn!

Dpn (n − 1)! (2n − 1)(n − 1)!

Dpk ,ln (k l)

(n+k +l)(n+2k −1)!(n+2l−1)!

2k +l−1n!k !l!

(2n+k +l−1)(n+k +l)(n+2k −1)!(n+2l−1)!

2k +l−1n!k !l!

Dpk ,k n

(n+2k )(n+2k −1)!2

22k n!k !2(2n+2k −1)(n+2k )(n+2k −1)!2

22k n!k !2

Table 8.1.1

§8.2 ROOTED MAP ON GENUS UNDERLYING A GRAPH

8.2.1 Rooted Map Polynomial. For a graph G with maximum valency ≥ 3, assume

that r i(G),r i(G), i ≥ 0 are respectively the numbers of rooted maps underlying graphG on orientable surface of genus γ (G) + i − 1 or on non-orientable surface of genusγ (G)+ i−1, where γ (G) andγ (G) denote the minimum orientable genus and the minimum

non-orientable genus of G, respectively. The rooted orientable map polynomial r [G]( x) ,

rooted non-orientable map polynomialr [G]( x) and rooted total map polynomial R[G]( x)

on genus are defined by

r [G]( x) =i≥0

r i(G) xi,

r [G]( x) = i≥0 r i(G) xi

and

R[G]( x) =i≥0

r i(G) xi +i≥1

r i(G) x−i.

We have known that the total number of orientable embeddings of G is

d ∈ D(G)(d − 1)!

and non-orientable embeddings is (2 β(G) −1)

d ∈ D(G)(d −1)! by Corollary 5.1.2, where D(G)



Sec.8.2 Rooted Maps on Genus Underlying a Graph 277

is its valency sequence. Similarly, let gi(G) and gi(G), i ≥ 0 respectively be the number

of embeddings of G on the orientable surface with genus γ (G) + i − 1 and on the non-

orientable surface with genus

γ (G) + i − 1. The orientable genus polynomial g[G]( x),

non-orientable genus polynomialg[G]( x) and total genus polynomial G[G]( x) of graph Gare defined respectively by

g[G]( x) =i≥0

gi(G) xi,

g[G]( x) =i≥0

gi(G) xi

and

G[G]( x) =i≥0

gi(G) xi +i≥1

gi(G) x−i.

All these polynomials r [G]( x), r [G]( x), R[G]( x) and g[G]( x), g[G]( x), G[G]( x) are finiteby properties of G on surfaces, for example, Theorem 5.1.2.

We establish relations between r [G]( x) and g[G]( x), r [G]( x) and g[G]( x), R[G]( x)

and G[G]( x) in the following result.

Theorem 8.2.1 For a connected graph G,

|Aut 12

G| r[G](x) = 2ε(G) g[G](x),

|Aut 12

G|

r[G](x) = 2ε(G)

g[G](x)

and

|Aut 12

G| R[G](x) = 2ε(G) G(x).

Proof For an integer k , denotes by Mk (G, S ) all the non-isomorphic maps on an

orientable surface S with genus γ (G) + k − 1. According to the Corollary 8.1.1, we know

that

r k (G) =

M ∈Mk (G,S )

4ε( M )

|Aut M |

=4ε(G)

|Aut 12

G × α | M ∈Mk (G,S )

|Aut 1

2

G

× α

||Aut M | .

Since |Aut 12

G × α | = |(Aut 12

G × α) M || M Aut 1

2G×α| and |(Aut 1

2G × α) M | = |Aut M |,

we know that

r k (G) =4ε(G)

|Aut 12

G × α |

M ∈Mk (G,S )

| M Aut 1

2G×α| =

2ε(G)gk (G)

|Aut 12

G| .




Consequently,

|Aut 12

G| r[G](x) = |Aut 12

G|i≥0

ri(G)xi

= i≥0

|Aut 12 G|ri(G)x

i

=i≥0

2ε(G)gi(G) xi = 2ε(G) g[G]( x).

Similarly, let Mk (G,S ) be all non-isomorphic maps on an non-orientable surface Swith genusγ (G) + k − 1. Similar to the orientable case, we get that

r k (G) =4ε(G)

|Aut 12

G × α |

M ∈

Mk (G,

S )

|Aut 12

G × α ||Aut M |

=

4ε(G)

|Aut 12

G × α | M ∈Mk (G,S )

| M

Aut 1

2

G

×α

|

=2ε(G)gk (G)

|Aut 12

G| .

Whence,

|Aut 12

G|r[G](x) =i≥0

|Aut 12

G|ri(G)xi

=i≥0

2ε(G)

gi(G) xi = 2ε(G)

g[G]( x).

Notice that

R[G]( x) =i≥0

r i(G) xi +i≥1

r i(G) x−i

and

G[G]( x) =i≥0

gi(G) xi +i≥1

gi(G) x−i.

We also get that

r k (G) =2ε(G)gk (G)

|Aut 12

G| and

r k (G) =

2ε(G)

gk (G)

|Aut 12

G|

for integers k ≥ 0. Therefore, we get that

|Aut 12

G| R[G](x) = |Aut 12

G|(i≥0

ri(G)xi +i≥1

ri(G)x−i)

=i≥0

|Aut 12

G|ri(G)xi +i≥1

|Aut 12

G|ri(G)x−i

=i≥0

2ε(G)gi(G) xi +i≥0

2ε(G)gi(G) x−i = 2ε(G) G[G]( x).





Corollary 8.2.1 Let G be a graph and s ≥ 0 an integer. If r s(G) and gs(G) are the numbers

of rooted maps and embeddings on a locally orientable surface of genus s underlying

graph G, respectively. Then

r s(G) =2ε(G)gs(G)

|Aut 12

G| .

8.2.2 Rooted Map Sequence. Corollary 8.2.1 can be used to find the implicit relations

among r [G]( x), r [G]( x) or R[G]( x) if the implicit relations among g[G]( x), g[G]( x) or

G[G]( x) are known, and vice via.

Denote the variable vector ( x1, x2, · · ·) by x¯

,

r¯(G) = (· · · ,r 2(G),r 1(G), r 0(G), r 1(G), r 2(G), · · ·),

g¯

(G) = (· · · ,g2(G),g1(G), g0(G), g1(G), g2(G), · · ·).

We call r¯(G) and g

¯(G) the rooted map sequence and the embedding sequence of graph G,

respectively.

Define a function F (x¯

, y¯

) to be y-linear if it can be represented as the following form

F (x¯

, y¯

) = f ( x1, x2, · · ·) + h( x1, x2, · · ·)i∈ I

yi + l( x1, x2, · · ·)Λ∈O

Λ(y¯

),

where I denotes a subset of index and O a set of linear operators. Notice that f ( x1, x2, · · ·) =

F (x¯

, 0¯

), where 0¯

= (0, 0, · · ·). We get the following general result.

Theorem 8.2.2 Let G be a graph family and H ⊆ G. If their embedding sequences

g¯

(G), G ∈ H satisfy the equation

F H ( x¯

, g¯

(G)) = 0, (4.1)

then the rooted map sequences r ¯ (G), G ∈ H satisfy the equation

F H ( x¯

,|Aut 1

2G|

2ε(G)r ¯

(G)) = 0,

and vice via, if the rooted map sequences r ¯


F H ( x¯

, r ¯

(G)) = 0, (4.2)




then the embedding sequences g¯


F H ( x¯

,2ε(G)

|Aut 12

G|g¯ (G)) = 0.

Furthermore, assume the function F ( x¯

, y¯

) is y-linear and 2ε(G)

|Aut 12

G| , G ∈ H is a constant.

If the embedding sequences g¯

(G), G ∈ H satisfy equation (4.1) , then

F ⋄H ( x¯ , r ¯

(G)) = 0,

where F ⋄H ( x¯ , y¯

) = F ( x¯

, y¯

) + (2ε(G)

|Aut 12

G| − 1)F ( x¯

, 0¯

) and vice via, if the rooted map sequences

g¯

(G), G ∈ H satisfy equation (4.2) , then

F ⋆H ( x , g (G)) = 0.

where F ⋆H = F ( x¯

, y¯

) + (|Aut 1

2G|

2ε(Γ)− 1)F ( x

¯ , 0¯

).

Proof According to the Corollary 8.2.1, for any integer s ≥ o and G ∈ H , we know

that

r s(G) =2ε(G)

|Aut 12

G| gs(G)

and

gs(G) =

|Aut 12

G|2ε(G)

r s(G).

Therefore, if the embedding sequences g¯

(G), G ∈ H satisfy equation (4.1), then

F H (x¯

,|Aut 1

2G|

2ε(G)r¯(G)) = 0,

and vice via, if the rooted map sequences r¯(G), G ∈ H satisfy equation (4.2), then

F H (x¯

,2ε(G)

|Aut 12

G|g¯(G)) = 0.

Now assume that F H

(x¯

, y¯

) is a y-linear function with a form

F H (x¯

, y¯

) = f ( x1, x2, · · ·) + h( x1, x2, · · ·)

i∈ I

yi + l( x1, x2, · · ·)Λ∈O

Λ(y¯

),

where O is a set of linear operators. If F H (x¯

, g¯

(G)) = 0, that is

f ( x1, x2, · · ·) + h( x1, x2, · · ·)

i∈ I , G∈H gi(G) + l( x1, x2, · · ·)

Λ∈O, G∈H

Λ(g¯

(G)) = 0,




we get that

f ( x1, x2, · · ·) + h( x1, x2, · · ·)

i

∈ I , G

∈H

|Aut 12

G|2ε(G)

r i(G)

+ l( x1, x2, · · ·)

Λ∈O, G∈H Λ(

|Aut 12

G|2ε(G)

r¯(G)) = 0.

Since Λ ∈ O is a linear operator and2ε(G)

|Aut 12

G| , G ∈ H is a constant, we also have

f ( x1, x2, · · ·) +|Aut 1

2G|

2ε(G)h( x1, x2, · · ·)

i∈ I , G∈H

r i(G)

+

|Aut 12

G|2ε(G) l( x1, x2, · · ·)

Λ∈O, G∈H Λ(r(G)) = 0,

that is,

2ε(G)

|Aut 12

G| f ( x1, x2, · · ·) + h( x1, x2, · · ·)

i∈ I , G∈H r i(G) + l( x1, x2, · · ·)

Λ∈O, G∈H

Λ(r¯(G)) = 0.

Consequently, we get that

F ⋄H (x¯

, r¯(G)) = 0.

Similarly, if F H (x

¯, r

¯(G)) = 0,

we can also get that

F ⋆H (x¯

, g¯

(G)) = 0.


Corollary 8.2.2 Let G be a graph family and H ⊆ G. If the embedding sequences g¯

(G)

of graph G ∈ G satisfy a recursive relation

i∈ J , G∈H

a(i, G)gi(G) = 0,

where J is the set of index, then the rooted map sequences r ¯

(G) satisfy a recursive relation

i∈ J , G∈H

a(i, G)|Aut 12

G|2ε(G)

r i(G) = 0,




and vice via.

A typical example of Corollary 8.2.2 is the graph family bouquets Bn, n ≥ 1. Notice

that the following recursive relation for the number gm(n) of embeddings of a bouquet Bn

on an orientable surface with genus m for n ≥ 2 was found in [GrF2].

(n + 1)gm(n) = 4(2n − 1)(2n − 3)(n − 1)2(n − 2)gm−1(n − 2)

+ 4(2n − 1)(n − 1)gm(n − 1)

with boundary conditions

gm(n) = 0 if m ≤ 0 or n ≤ 0;

g0(0) = g0(1) = 1 and gm(0) = gm(1) = 0 for m ≥ 0;

g0(2) = 4, g1(2) = 2, gm(2) = 0 for m ≥ 1.

Since |Aut 12 Bn| = 2nn!, we get a recursive relation for the number r m(n) of rooted

maps on an orientable surface of genus m underlying graph Bn by Corollary 8.2.2 follow-

ing.

(n2 − 1)(n − 2)r m(n) = (2n − 1)(2n − 3)(n − 1)2(n − 2)r m−1(n − 2)

+ 2(2n − 1)(n − 1)(n − 2)r m(n − 1)

with the boundary conditions r m(n) = 0 if m ≤ 0 or n ≤ 0;r 0(0) = r 0(1) = 1 and r m(0) = r m(1) = 0 for m ≥ 0;

r 0(2) = 2, r 1(2) = 1, gm(2) = 0 for m ≥ 1.

Corollary 8.2.3 Let G be a graph family and H ⊆ G. If the embedding sequences

g¯

(G), G ∈ G satisfy an operator equationΛ∈O, G∈H

Λ(g¯

(G)) = 0,

where O denotes a set of linear operators, then the rooted map sequences r ¯ (G), G ∈ H satisfy an operator equation

Λ∈O, G∈H

Λ(|Aut 1

2G|

2ε(G)r ¯

(G)) = 0

and vice via.




Let θ = (θ 1, θ 2, · · · , θ k ) ⊢ 2n, i.e.,k

j=1

θ j = 2n with positive integers θ j. Kwak and

Shim introduced three linear operators Γ, Θ and ∆ to find the total genus polynomial of

bouquets Bn, n

≥1 in [KwS1] defined as follows.

Denotes by zθ and z−1θ = 1/ zθ the multivariate monomials

k i=1

zθ i and 1/k

i=1

zθ i , where

θ = (θ 1, θ 2, · · · , θ k ) ⊢ 2n. Then the linear operators Γ, Θ and ∆ are defined respectively by

Γ( z±1θ ) =

k j=1

θ jl=0

θ j( z1+l zθ j+1−l

zθ j

) zθ ±1,

Θ( z±1θ ) =

k j=1

(θ 2 j + θ j)( zθ j+2 zθ

zθ j

)−1

and

∆( z±1θ ) =

1≤i< j≤k

2θ iθ j[( zθ j+θ i+2

zθ j zθ i

) zθ ±1 + ( zθ j+θ i +2

zθ j zθ i

) zθ −1].

Denote by i[ Bn]( z j) the sum of all monomial zθ or 1/ zθ taken over all embeddings of Bn

into an orientable or non-orientable surface, that is

i[ Bn]( z j) =θ ⊢2n

iθ ( Bn) zθ +θ ⊢2n

iθ ( Bn) z−1θ ,

where, iθ ( Bn) and iθ ( Bn) denote the number of embeddings of Bn into orientable and non-

orientable surface of region type θ . They found that

i[ Bn+1]( z j) = (Γ + Θ + ∆)i[ Bn]( z j) = (Γ + Θ + ∆)n( 1 z2

+ z21).

and

G[ Bn+1]( x) = (Γ + Θ + ∆)n(1

z2

+ z21)| z j = x f o r j≥1 and (C ∗),

where, (C ∗) denotes the condition

(C ∗): replacing the power 1 + n − 2i of x by i if i ≥ 0 and −(1 + n + i) by −i if i ≤ 0.

Notice that |Aut 12 Bn|

2ε( Bn)

=2nn!

2n

= 2n−1(n

−1)!

and Γ, Θ, ∆ are linear. By Corollary 8.2.3 we know that

R[ Bn+1]( x) =(Γ + Θ + ∆)i[ Bn]( z j)

2nn!| z j= x f o r j≥1 and (C ∗)

=(Γ + Θ + ∆)n( 1

z2+ z2

1)

nk =1

2k k !

| z j = x f o r j≥1 and (C ∗).





R[ B1]( x) = x +1

x;

R[ B2]( x) = 2 + x +

5

x +

4

x2 ;

R[ B3]( x) =41

x3+

42

x2+

22

x+ 5 + 10 x

and

R[ B4]( x) =488

x4+

690

x3+

304

x2+

93

x+ 14 + 70 x + 21 x2.

§8.3 A SCHEME FOR ENUMERATING MAPS UNDERLYING A GRAPH

For a given graph G, denoted by EO(G), E N (G) and E L(G) the sets of embeddings of G

on orientable surfaces, non-orientable surfaces and on locally orientable surfaces, respec-

tively. For determining the number of non-equivalent embeddings of a graph on sur-

faces and maps underlying a graph, another form of the Theorem 5.3.3 by group action is

needed, which is restated as follows.

Theorem 8.3.1 Let M 1 = (X α,β, P1) and M 2 = (X α,β, P2) be two maps underlying

graph G, then

(1) M 1, M 2 are equivalent if and only if M 1, M 2 are in one orbit of Aut 12 G action on

X 12

(G);

(2) M 1, M 2 are isomorphic if and only if M 1, M 2 are in one orbit of Aut 12

G × αaction on X α,β.

Now we can established a scheme for enumerating the number of non-isomorphic

maps and non-equivalent embeddings of a graph on surfaces by applying the well-known

Burnside Lemma, i.e., Theorem 2.1.3 in the following.

Theorem 8.3.2 For a graph G, let E ⊂ E

L(G) , then the numbers n(E

, G) and η(E

, G) of

non-isomorphic maps and non-equivalent embeddings in E are respective

n(E, G) =1

2|Aut 12

G|

g∈Aut 12

G

|Φ1(g)|,

η(E, G) =1

|Aut 12

G|

g∈Aut 12

G

|Φ2(g)|,



Sec.8.3 A Scheme for Enumerating Maps Underlying a Graph 285

where, Φ1(g) = P |P ∈ E and P g = P or Pgα = P , Φ2(g) = P |P ∈ E and

P g = P.

Proof Define the group

H = Aut 1

2G

× α

. Then by the Burnside Lemma and the

Theorem 8.3.1, we get that

n(E, G) =1

|H|g∈H

|Φ1(g)|,

where, Φ1(g) = P |P ∈ E and P g = P. Now |H| = 2|Aut 12

G|. Notice that if Pg = P ,

then Pgα P , and if Pgα = P, then P g

P. Whence, Φ1(g)

Φ1(gα) = ∅. We

have that

n(

E, G) =

1

2|Aut12 G| g∈Aut 1

2G |

Φ1(g)

|,

where Φ1(g) = P |P ∈ E and Pg = P or Pgα = P.

Similarly,

η(E, G) =1

|Aut 12

G|

g∈Aut 12

G

|Φ2(g)|,

where, Φ2(g) = P |P ∈ E and Pg = P.

From Theorem 8.3.2, we get results following.

Corollary 8.3.1 The numbers nO(G), n N (G) and n L(G) of non-isomorphic orientable

maps, non-orientable maps and locally orientable maps underlying a graph G are re-

spectively

nO(G) =1

2|Aut 12

G|

g∈Aut 12

G

|ΦO1 (g)|; (8.3.1)

n N (G) =1

2|Aut 1

2

G| g∈Aut 1

2G |

Φ N 1 (g)

|; (8.3.2)

n L(G) =1

2|Aut 12

G|

g∈Aut 12

G

|Φ L1 (g)|, (8.3.3)

where, ΦO1 (g) = P |P ∈ EO(G) and Pg = P or Pgα = P , Φ N

1 (g) = P |P ∈ E N (G)

and P g = P or Pgα = P , Φ L1 (g) = P |P ∈ E L(G) and Pg = P or Pgα = P.




Corollary 8.3.2 The numbers ηO(G), η N (G) and η L(G) of non-equivalent embeddings of

graph G on orientable ,non-orientable and locally orientable surfaces are respectively

ηO(G) =1

|Aut 12 G| g∈Aut 1

2G |

ΦO

2(g)

|; (8.3.4)

η N (G) =1

|Aut 12

G|

g∈Aut 12

G

|Φ N 2 (g)|; (8.3.5)

η L(G) =1

|Aut 12

G|

g∈Aut 12

G

|Φ L2 (g)|, (8.3.6)

where, ΦO2

(g) = P |P ∈ EO(G) and Pg = P , Φ N 2

(g) = P |P ∈ E N (G) and Pg = P ,

Φ L2 (g) = P |P ∈ E

L

(G) and Pg

= P.

For a simple graph G, since Aut 12

G = AutG by Theorem 3.4.1, the formula (8.3.4)

is just the scheme used for counting the non-equivalent embeddings of a complete graph,

a complete bipartite graph in references [MRW1], [Mul1]. For an asymmetric graph G,

that is, Aut 12

G = id X 12

(G), we get the numbers of non-isomorphic maps and non-equivalent

embeddings underlying graph G by the Corollaries 8.3.1 and 8.3.2 following.

Theorem 8.3.3 The numbers nO(G) , n N (G) and n L(G) of non-isomorphic maps on ori-

entable, non-orientable surfaces or locally orientable surfaces underlying an asymmetric

graph G are respectively

nO(G) =

v∈V (G)

( ρ(v) − 1)!

2,

n L(G) = 2 β(G)−1

v∈V (G)

( ρ(v) − 1)!

and

n N (G) = (2 β(G)−1 − 1

2)

v∈V (G)

( ρ(v) − 1)!,

where, β(G) is the Betti number of graph G.

The numbers ηO(G) , η N (G) and η L(G) of non-equivalent embeddings underlying an

asymmetric graph G are respectively

ηO(G) =

v∈V (G)

( ρ(v) − 1)!,



Sec.8.4 The Enumeration of Complete Maps on Surfaces 287

η L(G) = 2 β(G)

v∈V (G)

( ρ(v) − 1)!

and

η N (G) = (2 β(G)

−1) v∈V (G)

( ρ(v)−

1)!.

All these formulae are useful for enumerating non-isomorphic maps underlying a com-

plete graph, semi-regular graph or a bouquet on surfaces in sections following.

§8.4 THE ENUMERATION OF COMPLETE MAPS ON SURFACES

We first consider a permutation with its stabilizer. A permutation with the following form

( x1, x2, · · · , xn)(α xn, α x2, · · · , α x1) is called a permutation pair . The following result isobvious.

Lemma 8.4.1 Let g be a permutation on set Ω = x1, x2, · · · , xn such that gα = αg. If

g( x1, x2, · · · , xn)(α xn, α xn−1, · · · , α x1)g−1 = ( x1, x2, · · · , xn)(α xn, α xn−1, · · · , α x1),

then

g = ( x1, x2, · · · , xn)k

and if

gα( x1, x2, · · · , xn)(α xn, α xn−1, · · · , α x1)(gα)−1 = ( x1, x2, · · · , xn)(α xn, α xn−1, · · · , α x1),

then

gα = (α xn, α xn−1, · · · , α x1)k

for some integer k , 1 ≤ k ≤ n.

Lemma 8.4.2 For each permutation g, g ∈ E[k

nk ]

satisfying gα = αg on set Ω =

x1, x2,

· · ·, xn

, the number of stable permutation pairs in Ω under the action of g or

gα is2φ(k )(n − 1)!

|E[k

nk ]| ,

where φ(k ) denotes the Euler function.

Proof Denote the number of stable pair permutations under the action of g or gα

by n(g) and C the set of pair permutations. Define the set A = (g, C )|g ∈ E[k

nk ]

, C ∈




C and C g = C or C gα = C . Clearly, for ∀g1, g2 ∈ E[k

nk ]

, we have n(g1) = n(g2).

Whence, we get that

| A

|=

|E[k

nk ]

|n(g). (8.4.1)

On the other hand, by the Lemma 8.4.1, for any permutation pair C = ( x1, x2, · · · , xn)

(α xn, α xn−1, · · · , α x1), since C is stable under the action of g, there must be g = ( x1, x2, · · · , xn)l

or gα = (α xn, α xn−1, · · · , α x1)l, where l = s nk

, 1 ≤ s ≤ k and (s, k ) = 1. Therefore, there

are 2φ(k ) permutations in E[k

nk ]

acting on it stable. Whence, we also have

| A| = 2φ(k )|C|. (8.4.2)

Combining (8.4.1) with (.4.2), we get that

n(g) =

2φ(k )

|C||E[k nk ]| =

2φ(k )(n

−1)!

|E[k nk ]| .

Now we can enumerate the unrooted complete maps on surfaces.

Theorem 8.4.1 The number n L(K n) of complete maps of order n ≥ 5 on surfaces is

n L(K n) =1

2(

k |n+

k |n,k ≡0(mod 2)

)2α(n,k )(n − 2)!

nk

k nk ( n

k )!

+

k |(n−1),k 1

φ(k )2 β(n,k )(n − 2)!n−1

k

n − 1,

where,

α(n, k ) = n(n − 3)

2k

, if k

≡1(mod 2);

n(n − 2)

2k , if k ≡ 0(mod 2),

and

β(n, k ) =

(n − 1)(n − 2)

2k , if k ≡ 1(mod 2);

(n − 1)(n − 3)

2k , if k ≡ 0(mod 2).

and n L(K 4) = 11.

Proof According to formula (8.3.3) in Corollary 8.3.1 and Theorem 7.2.1 for n ≥ 5,

we know that

n L(K n) =1

2|AutK n|×

g1∈E[k

nk ]

|Φ(g1)| +

g2∈E[(2s)

n2s ]

|Φ(g2α)| +

h∈E[1,k

n−1k ]

|Φ(h)|

=

1

2n!×

k |n|E

[k nk ]||Φ(g1)| +

l|n,l≡0(mod 2)

|E[l

nl ]||Φ(g2α)| +

l|(n−1)

|E[1,l

n−1l ]

||Φ(h)| ,




where, g1 ∈ E[k

nk ]

, g2 ∈ E[l

nl ]

and h ∈ E[1,k

n−1k ]

are three chosen elements.

Without loss of generality, we assume that an element g, g ∈ E[k

nk ]

has the following

cycle decomposition.

g = (1, 2, · · · , k ) (k + 1, k + 2, · · · , 2k ) · · · n

k − 1

k + 1,n

k − 1

k + 2, · · · , n

and

P =

1×

2,

where

1

=

1i21 , 1i31 , · · · , 1in1

2i12 , 2i32 , · · · , 2in2

· · ·

ni1n , ni2n , · · · , ni(n−1)n

,

and 2

= α

1

−1

α−1,

being a complete map which is stable under the action of g, where si j ∈ k +, k − |k =

1, 2, · · · , n.

Notice that the quadricells adjacent to the vertex 1 can make 2n−2(n − 2)! diff erent

pair permutations and for each chosen pair permutation, the pair permutations adjacent to

the vertices 2, 3, · · · , k are uniquely determined since P is stable under the action of g.

Similarly, for each pair permutation adjacent to the vertex k + 1, 2k + 1, · · · , n

k − 1 k +1, the pair permutations adjacent to k + 2, k + 3, · · · , 2k , and 2k + 2, 2k + 3, · · · , 3k ,· · ·,and

n

k − 1

k + 2,

n

k − 1

k + 3, · · · , n are also uniquely determined because P is stable

under the action of g.

Now for an orientable embedding M 1 of K n, all the induced embeddings by ex-

changing two sides of some edges and retaining the others unchanged in M 1 are the same

as M 1 by the definition of maps. Whence, the number of diff erent stable embeddings

under the action of g gotten by exchanging x and α x in M 1 for x ∈ U , U ⊂ X β, where

X β= x∈ E (K n)

x, β x , is 2

g(ε)

−nk

, whereg

(ε

) is the number of orbits of E

(K

n) under the action

of g and we substractn

k because we can chosen 12+, k + 11+, 2k + 11+, · · · , n − k + 11+ first

in our enumeration.

Notice that the length of each orbit under the action of g is k for ∀ x ∈ E (K n) if k is

odd and isk

2for x = ii+ k

2 , i = 1, k + 1, · · · , n − k + 1, or k for all other edges if k is even.

Therefore, we get that




g(ε) =

ε(K n)

k , if k ≡ 1(mod2);

ε(K n)

−n2

k , if k ≡ 0(mod2).

Whence, we have that

α(n, k ) = g(ε) − n

k =

n(n − 3)

2k , if k ≡ 1(mod2);

n(n − 2)

2k , if k ≡ 0(mod2),

and

|Φ(g)

|= 2α(n,k )(n

−2)!

nk , (8.4.3)

Similarly, if k ≡ 0(mod2), we get also that

|Φ(gα)| = 2α(n,k ) (n − 2)!nk (8.4.4)

for an chosen element g,g ∈ E[k

nk ]

.

Now for ∀h ∈ E[1,k

n−1k ]

, without loss of generality, we assume that h = (1, 2, · · · , k )

(k + 1, k + 2, · · · , 2k ) · · ·

n − 1

k − 1

k + 1,

n − 1

k − 1

k + 2, · · · , (n − 1)

(n). Then the

above statement is also true for the complete graph K n−

1 with the vertices 1, 2,

· · ·, n

−1.

Notice that the quadricells n1+, n2+, · · · , nn−1+ can be chosen first in our enumeration and

they are not belong to the graph K n−1. According to the Lemma 8.4.2, we get that

|Φ(h)| = 2 β(n,k )(n − 2)!n−1

k × 2φ(k )(n − 2)!

|E[1,k

n−1k ]

| , (8.4.5)

Where

β(n, k ) = h(ε) =

ε(K n−1)

k − n − 1

k =

(n − 1)(n − 4)

2k , if k ≡ 1(mod2);

ε(K n−1)

k −n

−1

k =

(n

−1)(n

−3)

2k , if k ≡ 0(mod2).

Combining (8.4.3) − (8.4.5), we get that

n L(K n) =1

2n!× (

k |n|E

[k nk ]||Φ(g0)| +

l|n,l≡0(mod 2)

|E[l

nl ]||Φ(g1α)|

+

l|(n−1)

|E[1,l

n−1l ]

||Φ(h)|)




=1

2n!× (

k |n

n!2α(n,k )(n − 2)!nk

k nk ( n

k )!

+

k |n,k ≡0(mod 2)

n!2α(n,k )(n − 2)!nk

k nk ( n

k )!

+ k |(n−1),k 1

n!

k n

−1

k (n−

1k )! ×

2φ(k )(n − 2)!2 β(n,k )(n − 2)!n−1

k

(n−

1)!

k n−1

k ( n−1k

)!

)

=1

2(

k |n+

k |n,k ≡0(mod 2)

)2α(n,k )(n − 2)!

nk

k nk ( n

k )!

+

k |(n−1),k 1

φ(k )2 β(n,k )(n − 2)!n−1

k

n − 1.

For n = 4, similar calculation shows that n L(K 4) = 11 by consider the fixing set of

permutations in E[s

4s ]

, E[1,s

3s ]

, E[(2s)

42s ]

, αE[(2s)

42s ]

and αE[1,1,2].

For the orientable case, we get the number nO(K n) of orientable complete maps of

order n as follows.

Theorem 8.4.2 The number nO((K n) of complete maps of order n ≥ 5 on orientable

surfaces is

nO(K n) =1

2(

k |n+

k |n,k ≡0(mod 2)

)(n − 2)!

nk

k nk ( n

k )!

+

k |(n−1),k 1

φ(k )(n − 2)!n−1

k

n − 1.

and n(K 4) = 3.

Proof According to the algebraic representation of map, a map M = (X α,β, P) is

orientable if and only if for ∀ x ∈ X α,β, x and αβ x are in a same orbit of X α,β under the

action of the group Ψ I =

αβ,

P. Now applying (8.3.1) in Corollary 8.3.1 and Theorem

7.2.1, similar to the proof of Theorem 8.4.1, we get the number nO(K n) for n ≥ 5 to be

nO(K n) =1

2(

k |n+

k |n,k ≡0(mod 2)

)(n − 2)!

nk

k nk ( n

k )!

+

k |(n−1),k 1

φ(k )(n − 2)!n−1

k

n − 1.

and for the complete graph K 4, calculation shows that n(K 4) = 3.

Notice that nO(K n) + n N (K n) = n L(K n). Therefore, we get also the number n N (K n)

of complete maps of order n on non-orientable surfaces by Theorems 8.4.1 and 8.4.2

following.

Theorem 8.4.3 The number n N (K n) of complete maps of order n, n ≥ 5 on non-orientable

surfaces is

n N (K n) =1

2(

k |n+

k |n,k ≡0(mod 2)

)(2α(n,k ) − 1)(n − 2)!

nk

k nk ( n

k )!

+

k |(n−1),k 1

φ(k )(2 β(n,k ) − 1)(n − 2)!n−1

k

n − 1,




and n N (K 4) = 8. Where, α(n, k ) and β(n, k ) are the same as in Theorem 8.4.1.

For n = 5, calculation shows that n L(K 5) = 1080 and nO(K 5) = 45 by Theorems 8.4.1

and 8.4.2. For n = 4, there are 3 orientable complete maps and 8 non-orientable complete

maps shown in the Fig.8.4.1.

1

1

2 2

1

1

2 2

¹

¹

¹

¹

1

1

1

1

22

1

1

22

¹

¹

1

1

2 2

1

2

1

2

11

2

23

3

11

2

23

3

3’

3’ 1 1

2

2

3

33’

3’

«

Í

c1

Ã

¹

c1

c2

c2

c3

c3

¹

¹

c3

c3

c1

c1

c2

c2

c3

c3

c1

c1

c2

c2

Fig.8.4.1

Now consider the action of orientation-preserving automorphisms of complete maps,

determined in Theorem 7.2.1 on all orientable embeddings of a complete graph of order

n. Similar to the proof of the Theorem 8.4.2, we can get the number of non-equivalent

embeddings of a complete graph of order n, which has been found in [Mao1] and it is the

same gotten by Mull et al. in [MRW1].



Sec.8.5 The Enumeration of Maps Underlying Semi-Regular Graph 293

§8.5 THE ENUMERATION OF MAPS UNDERLYING A SEMI-REGULAR GRAPH

8.5.1 Crosscap Map Group. For a given map M = (X α,β, P), its crosscap map group

is defined to be

T :=< τ|∀ x ∈ X , τ = ( x, α x) >,

where, X = E (G). Consider the action of T on M . For ∀θ ∈ T , we define

M θ := (X α,β, θ Pθ −1);

M T := M θ |∀θ ∈ T.

Then we have the following lemmas.

Lemma 8.5.1 Let G be a connected graph. Then for ∀

M ∈ E

T (G) , there exists an element

τ, τ ∈ T and an embedding M 0, M 0 ∈ EO(G) such that

M = M τ0 .

Lemma 8.5.2 For a connected graph G,

ET (G) = M τ| M ∈ EO(G), τ ∈ T.

We need to classify maps in ET (G). The following lemma is fundamental for this

objective.

Lemma 8.5.3 For maps M , M 1 ∈ EO(G) , if there exist g ∈ AutG and τ ∈ T such that

( M g)τ = M 1 , then there must be M 1 isomorphic to M and τ ∈ T M 1 , and moreover, if

M 1 = M, then g ∈ Aut M.

Proof We only need to prove that if M g = M τ1

, g ∈ AutG and τ ∈ T ,then τ ∈ T M 1 .

Assume that M = (X α,β, P), M 1 = (X α,β, P1), P = C αC −1,P1 = C 1αC −11 and τ = τS ,

where S ⊂ C 1. For ∀ x ∈ C , a direct calculation shows that

Pg

= · · · ( x, , g( y1), g( y2), · · · , g( yt ))(α x, αg( yt ), · · · , αg( y1)) · · · ;

P τ1 = · · · (τ x, τ z1, τ z2, · · · , τ zs)(ατ x, ατ zs, · · · , ατ z1) · · · , (8.5.1)

where

P = · · · ( x, x1, x2, · · · , xs)( y, y1, y2, · · · , yt ) · · · ;

P1 = · · · ( x, z1, z2, · · · , zs)(α x, α zs, · · · , α z1)




and g( y) = x, zi ∈ v x, i ∈ 1, 2, · · · , sSince g ∈ AutG, we know that

y, y1,

· · ·, yt

g =

x, x1,

· · ·, xs

= x, z1, · · · , zs (8.5.2)

and t = s. Now we consider two cases.

Case 1. x S .

In this case, we get that P τ1 = · · · ( x, τ z1, τ z2, · · · , τ zs)(α x, ατ zs, · · · , ατ z1) · · · , from

(8.5.2). Since Pg = P τS

1 , we get that g( y1) = τ z1, g( y2) = τ z2, · · · , g( ys) = τ zs. According

to (8.5.2), we know that g( y1) = z1, g( y2) = z2, · · · , g( ys) = zs. Therefore, z1 S , z2

S , · · · , zs S , that is v x S .

Case 2. x ∈ S .

In this case, we have that Pτ1 = · · · (α x, τ z1, τ z2, · · · , τ zs)( x, ατ zs, · · · , ατ z1) · · · , Be-

cause of Pg = PτS

1 , we get that g( y1) = ατ zs, g( y2) = ατ zs−1, · · · , g( ys) = ατ z1. Ac-

cording to (8.5.2) again, we find that g( y1) = zs, g( y2) = zs−1, · · · , g( ys) = z1. Whence,

z1 ∈ S , z2 ∈ S , · · · , zs ∈ S , that is v x ⊂ S .

Combining the discussion of Cases 1 and 2, we know that there exists a vertex subset

V 1 ⊂ V (G) such that V 1 = S . Whence τ ∈ T M 1 . Since M g = M τ1 = M 1, we get that M 1 is

isomorphic to M .

Now if M 1 = M , we also get that M g = M . Therefore, g ∈ Aut M

We get the following result by Lemmas 8.5.1 - 8.3.1.

Theorem 8.5.1 Let G be a connected graph. Then

(1) For ∀ M τS

1∈ M T

1, M τ R

2∈ M T

2, where M 1, M 2 ∈ EO(G) , if M

τS

1is isomorphic to

M τ R

2, then M 1 is also isomorphic to M 2.

(2) For a given M ∈ EO(G) , ∀ M τS , M τ R ∈ M T , there exists an isomorphism g such

that g : M τS

→M τ R if and only if g

∈AutM and τ R

∈τg−1(S )

· T M .

Proof (1) Assume g ia an isomorphism between M τS

1and M τ R

2, thus ( M

τS

1)g = M τ R

2.

Since

g−1τS g = g−1( x∈S

( x, α x))g = x∈S

(g−1 x, αg−1 x)

=

x∈g−1(S )

( x, α x) = τg−1(S ),




we get that τS g = gτg−1(S ). Whence,

( M g

1 )τg−1(S )

·τ−1 R = M 2.

According to Lemma 8.5.3, M 1 is isomorphic to M 2.

(2) Notice that there must be g ∈ AutG. Since ( M τS )g = M τ R , we find that

( M g)τ

g−1(S )·τ−1

R = M .

According to Lemma 8.5.3 again, we get that

g ∈ AutM and τ R ∈ τg−1(S )T M

.

On the other hand, if there exist τ

∈ T and g

∈Aut M such that τ R = τg−1(S )

·τ, then

( M τS )g = ( M g)τ

g−1(S ) = M τ

g−1 (S ) = M τ R .

Therefore, g is an isomorphism between M τS and M τ R .

8.5.2 Enumerating Semi-Regular Map. We enumerate maps underlying a semi-regular

graph on orientable or non-orientable surfaces.

Lemma 8.5.4 Let G = (V , E ) be a semi-regular graph. Then for ξ ∈ AutG

|ΦO( ξ )| = x∈T V

ξ

(d ( x)

o( ξ | N G( x)) − 1)!

and

|Φ L( ξ )| = 2|T E ξ

|−|T V ξ

|

x∈T V ξ

(d ( x)

o( ξ | N G( x))− 1)!,

where, T V ξ

, T E ξ

are the representations of orbits of ξ acting on V (G) and E (G) ,respectively

and ξ N G( x) the restriction of ξ to N G ( x).

Proof According to Theorem 8.5.1, we know that

ET (G) = Pτ|P ∈ EO(G), τ ∈ T

Notice that if M ξ = M , then M τ ξ = M τ. Now since AutG is semi-regular acting on E (G),

we can assume that

ξ |V (G) = (a, b, · · · , c) · · · (d , e, · · · , f ) · · · ( x, y, · · · , z)




and

ξ | E (G) = (e11, e12, · · · , e1l1) · · · (ei1, ei2, · · · , eili

) · · · (es1, es2, · · · , esls).

For a stable orientable embedding M 0 = ( E (G)α,β, P0) under the action of ξ , it is clear

that

|Φ( M T 0 , ξ )| = 2orb( ξ | E (G))−orb( ξ |V (G) ),

where orb( ξ ) E (G) and orb( ξ |V (G)) are the number of orbits of E (G), V (G) under the action

of ξ and we subtract orb( ξ |V (G)) because one of quadricells in vertices a, · · · , d , · · · , x can

be chosen first in our enumeration. Now since orb( ξ | E (G)) = |T E ξ | and orb( ξ |V (G)) = |T V

ξ |,

we get that

|Φ( M T 0 , ξ )| = 2|T E

ξ |−|T V

ξ |.

Notice that if the rotation of the quadricells adjacent to the vertex a has been given,then the rotations adjacent to the vertices b, · · · , c are uniquely determined if the cor-

respondence embedding is stable under the action of ξ . Similarly, if a rotation of the

quadricells adjacent to the vertices a, · · · , d , · · · , x have been given, then the map M =

( E (G)α,β, P) is uniquely determined if M is stable under the action of ξ . Since ξ | N G( x) is

semi-regular, for ∀ x ∈ V (G) we can assume that

ξ | N G( x) = ( x z1 , x z2 , · · · , x zs )( x zs+1 , x zs+2 , · · · , x z2s ) · · · ( x z(k −1)s+1 , x z(k −1)s+2, · · · , x zks ).


|ΦO( ξ )| =

x∈T V ξ

(d ( x)

o( ξ | N G( x))− 1)!.

According to the Corollary 8.3.1, we get enumeration results following.

Theorem 8.5.2 Let G be a semi-regular graph. Then the numbers of maps underlying the

graph G on orientable or non-orientable surfaces are respectively

nO(G) =1

|AutG|( ξ ∈AutG

λ( ξ ) x∈T V ξ

(d ( x)

o( ξ | N G( x)) −1)!

and

n N (G) =1

|AutG| ×

ξ ∈AutG

(2|T E

ξ |−|T V

ξ | − 1)λ( ξ )

x∈T V

ξ

(d ( x)

o( ξ | N G( x))− 1)!,

where λ( ξ ) = 1 if o( ξ ) ≡ 0(mod2) and 1

2 , otherwise.




Proof By the Corollary 8.3.1, we know that

nO(G) =1

2|Aut 12

G|

g∈Aut 12

G

|ΦO1 (g)|

and

n L(G) =1

2|Aut 12

G|

g∈Aut 12

G

|ΦT 1 (g)|.

According to the Theorem 7.3.4, all automorphisms of orientable maps underlying graph

G are respectively

g|Xα,β and αh|Xα,β , g, h ∈ AutG with o(h) ≡ 0(mod2).

and all the automorphisms of non-orientable maps underlying graph G are also

g|Xα,β and αh|Xα,β , g, h ∈ AutG with o(h) ≡ 0(mod2).

Whence, we get the number of orientable maps by the Lemma 8.5.4 as follows.

nO(G) =1

2|AutG|

g∈AutG

|ΦO1 (g)|

=1

2|AutG| (

ξ ∈AutG

x∈T V

ξ

(d ( x)

o( ξ | N G( x))− 1)!

+ ς ∈AutG,o(ς )≡0(mod2) x∈T V ς

(d ( x)

o(ς | N G( x)) −1)!)

=1

|AutG|(

ξ ∈AutG

λ( ξ )

x∈T V ξ

(d ( x)

o( ξ | N G( x))− 1)!.

Similarly, we enumerate maps underlying graph G on locally orientable surface by

(8.3.3) in Corollary 8.3.1 following.

n L(G) =1

2|AutG|

g∈AutG

|ΦT 1 (g)|

= 12|AutG|(|T E

ξ |−|T V

ξ |

ξ ∈AutG

x∈T V

ξ

( d ( x)o( ξ | N G( x))

− 1)!

+

ς ∈AutG,o(ς )≡0(mod2)

2|T E ς |−|T V

ς |

x∈T V ς

(d ( x)

o(ς | N G( x))− 1)!)

=1

|AutG|

ξ ∈AutG

λ( ξ )2|T E

ξ |−|T V

ξ |

x∈T V ξ

(d ( x)

o( ξ | N G( x))− 1)!.




Notice that nO(G) + n N (G) = n L(G). We get the number of maps on non-orientable

surfaces underlying graph G to be

n N (G) = n L(G)

−nO(G)

=1

|AutG| × ξ ∈AutG

(2|T E ξ

|−|T V ξ

| − 1)λ( ξ )

x∈T V ξ

(d ( x)

o( ξ | N G( x))− 1)!


Furthermore, if G is k -regular, we get a simple result for the numbers of maps on

orientable or non-orientable surfaces following.

Corollary 8.5.1 Let G be a k-regular semi-regular graph. Then the numbers of maps on

orientable or non-orientable surfaces underlying graph G are respectively

nO(G) =1

|AutG| × g∈AutG

λ(g)(k − 1)!|T V g |

and

n N (G) =1

|AutG| ×

g∈AutG

λ(g)(2|T E g |−|T V

g | − 1)(k − 1)!|T V g |,

where, λ( ξ ) = 1 if o( ξ ) ≡ 0(mod2) and 1

2 , otherwise.

Proof Notice that for ∀ ξ ∈ AutG, ξ is semi-regular acting on ordered pairs of adja-

cent vertices of G. Therefore, ξ is an orientation-preserving automorphism of map withunderlying graph of G.

Assume that

ξ V (G) = (a1, a2, · · · , as)(b1, b2, · · · , bs) · · · (c1, c2, · · · , cs).

It can be directly checked that for ∀e ∈ E (G),

|e<ξ> | = s ors

2.

The later is true only if s is an even number. Therefore, we have that

∀ x ∈ V (G), o( ξ N Γ( x)) = 1.

Whence, we get nO(G) and n N (G) by Theorem 8.5.2.

Similarly, if G = Cay( Z p : S ) for a prime p, we can also get closed formulas for the

number of maps underlying graph Γ.



Sec.8.6 The Enumeration of Bouquet on Surfaces 299

Corollary 8.5.2 Let G = Cay( Z p : S ) be a connected graph of prime order p with

( p − 1, |S |) = 2. Then

n

O

(G, M ) =

(

|S

| −1)! p + 2 p(

|S

| −1)!

p+12 + ( p

−1)(

|S

| −1)!

4 p

and

n N (G, M ) =(2

p|S |2

− p − 1)(|S | − 1)! p + 2(2 p|S |−2 p−2)

4 − 1) p(|S | − 1)! p+1

2

2 p

+(2

|S |−22 − 1)( p − 1)(|S | − 1)!

4 p.

Proof We calculate |T V g |, |T E

g | now. Since p is a prime number, there are p−1 elements

of degree p, p elements of degree 2 and one element of degree 1. Therefore, we know

that

|T V g | =

1, if o(g) = p p+1

2, if o(g) = 2

p, if o(g) = 1

and

|T E g | =

|S |2

, if o(g) = p p|S |

4, if o(g) = 2

p|S |2 , if o(g) = 1

Notice that AutG = D p and there are p elements order 2, one order 1 and p − 1 order p.

Whence, we have

nO(G, M ) =(|S | − 1)! p + 2 p(|S | − 1)!

p+12 + ( p − 1)(|S | − 1)!

4 p

and

n N (G, M ) =(2

p|S |2

− p − 1)(|S | − 1)! p + 2(2 p|S |−2 p−2)

4 − 1) p(|S | − 1)! p+1

2

2 p

+(2

|S |−22 − 1)( p − 1)(|S | − 1)!

4 p.

By Corollary 8.5.1.




§8.6 THE ENUMERATION OF A BOUQUET ON SURFACES

8.6.1 Cycle Index of Group. Let (Γ; ) be a group. Its cycle index of a group, denoted

by Z (Γ; s1, s2,· · ·

, sn) is defined by

Z (Γ; s1, s2, · · · , sn) =1

|G|g∈G

sλ1(g)

1s

λ2(g)

2· · · sλn(g)

n ,

where, λi(g) is the number of i-cycles in the cycle decomposition of g. For the symmetric

group S n, its cycle index is known to be

Z (S n; s1, s2, · · · , sn) =

λ1+2λ2 +···+k λk =n

sλ1

1sλ2

2· · · s

λk

k

1λ1 λ1!2λ2 λ2! · · · k λk λk !.

For example, we have that Z (S 2) =s2

1+s2

2

. By a result of Polya ( See [GrW1] for details),

we know that the cycle index of S n[S 2] is

Z (S n[S 2]; s1, s2, · · · , s2n) =1

2nn!

λ1+2λ2 +···+k λk =n

(s2

1+s2

2)λ1 (

s22

+s4

2)λ2 · · · (

s2k

+s2k

2)λk

1λ1 λ1!2λ2 λ2! · · · k λk λk !

8.6.2 Enumerating One-Vertex Map. For any integer k , k |2n, let J k be the conjugacy

class in S n[S 2] with each cycle in the decomposition of a permutation in J k being k -cycle.

According to Corollary 8.3.1, we need to determine the numbers |ΦO( ξ )| and |Φ L( ξ )| for

each automorphism of map underlying Bn.

Lemma 8.6.1 Let ξ =2n/k i=1

(C (i))(αC (i)α−1) ∈ J k be a cycle decomposition of ξ , where

C (i) = ( xi1, xi2, · · · , xik ) is a k-cycle. Then

(1) If k 2n, then

|ΦO( ξ )| = k 2nk (

2n

k − 1)!

and if k = 2n, then |ΦO( ξ )| = φ(2n).

(2) If k ≥ 3 and k 2n, then

|Φ L

( ξ )| = (2k )

2n

k −1

(

2n

k − 1)!

and

|Φ L( ξ )| = 2n(2n − 1)!

if ξ = ( x1)( x2) · · · ( xn)(α x1)(α x2) · · · (α xn)( β x1)( β x2) · · · ( β xn)(αβ x1)(αβ x2) · · · (αβ xn) , and

|Φ L( ξ )| = 1




if ξ = ( x1, αβ x1)( x2, αβ x2) · · · ( xn, αβ xn)(α x1, β x1)(α x2, β x2) · · · (α xn, β xn) , and

|Φ L( ξ )| =n!

(n − 2s)!s!

if ξ = ζ ; ε1, ε2, · · · , εn and ζ ∈ E[1n−2s ,2s] for some integer s, εi = (1, αβ) for 1 ≤ i ≤ s

and ε j = 1 for s + 1 ≤ j ≤ n, where E[1n−2s ,2s] denotes the conjugate class with the type

[1n−2s, 2s] in the symmetry group S n , and

|Φ L( ξ )| = φ(2n)

if ξ = θ ; ε1, ε2, · · · , εn and θ ∈ E[n1] and εi = 1 for 1 ≤ i ≤ n − 1 , εn = (1, αβ) , where φ(t )

is the Euler function.

Proof (1) Notice that for a representation of C (i), i = 1, 2, · · · ,2n

k , because the

group Pn, αβ is not transitive on X α,β, there is one and only one stable orientable map

Bn = (X α,β, Pn) with X = E ( Bn) and Pn = C (αC −1α−), where,

C = ( x11, x21, · · · , x 2nk

1, x21, x22, · · · , x 2nk

2, x1k , x2k , · · · , x 2nk

k ).

Counting ways for each possible order for C (i), i = 1, 2, · · · ,2n

k and diff erent representa-

tions for C (i), we know that

|ΦO( ξ )| = k 2nk (

2n

k − 1)!

for k 2n.

Now if k = 2n, then the permutation is itself a map underlying graph Bn. Whence,

its power is also an automorphism of this map. Therefore, we get that

|ΦO( ξ )| = φ(2n).

(2) For k ≥ 3 and k 2n, because the group Pn, αβ is transitive on X α,β or

not, we can interchange C (i) by αC (i)−1α−1 for each cycle not containing the quadricell

x11. Notice that we get the same map if the two sides of some edges are interchanged

altogether or not. Whence, we find that

|Φ L( ξ )| = 22nk

−1k 2nk

−1(2n

k − 1)! = (2k )

2nk

−1(2n

k − 1)!.

Now if ξ = ( x1, αβ x1)( x2, αβ x2) · · · ( xn, αβ xn)(α x1, β x1)(α x2, β x2) · · · (α xn, β xn), there

is one and only one stable map (X α,β, P1n ) under the action of ξ , where

P 1n = ( x1, x2, · · · , xn, αβ x1, αβ x2, · · · , αβ xn)(α x1, β xn, · · · , β x1, α xn, · · · , α x1),




which is orientable. Whence, |Φ L( ξ )| = |ΦO( ξ )| = 1.

If ξ = ( x1)( x2) · · · ( xn)(α x1)(α x2) · · · (α xn)( β x1)( β x2) · · · ( β xn)(αβ x1)(αβ x2) · · · (αβ xn),

we can interchange (αβ xi) with ( β xi) and obtain diff erent embeddings of Bn on surfaces.

Whence,|Φ L( ξ )| = 2n(2n − 1)!.

Now if ξ = (ζ ; ε1, ε2, · · · , εn) and ζ ∈ E[1n−2s,2s] for some integer s, εi = (1, αβ) for

1 ≤ i ≤ s and ε j = 1 for s + 1 ≤ j ≤ n, we can not interchange ( xi, αβ xi) with (α xi, β xi)

to get diff erent embeddings of Bn for it is just interchanging the two sides of one edge.


|Φ L( ξ )| =n!

1n−2s(n − 2s)!2s s!× 2s =

n!

(n − 2s)!s!.

For ξ = (θ ; ε1, ε2, · · · , εn), θ ∈ E[n1] and εi = 1 for 1 ≤ i ≤ n − 1, εn = (1, αβ), we can

not get diff erent embeddings of Bn by interchanging the two conjugate cycles. Whence,

we get that

|Φ L( ξ )| = |ΦO( ξ )| = φ(2n).


Now we enumerate maps on surfaces underlying graph Bn by Lemma 8.6.1.

Theorem 8.6.1 For an integer n ≥ 1 , the number nO( Bn) of maps on orientable surfaces

underlying graph Bn is

nO( Bn) =

k |2n,k 2n

k 2nk

−1(2n

k − 1)!

1

( 2nk

)!

∂2nk ( Z (S n[S 2]))

∂s2nk

k

|sk =0

+ φ(2n)∂( Z (S n[S 2]))

∂s2n

|s2n=0

Proof According to the formula (8.3.1) in Corollary 8.3.1, we know that

nO( Bn) =1

2

×2nn! ξ

∈S n[S 2]

×≺α≻

|ΦT ( ξ )|.

Since for ∀ ξ 1, ξ 2 ∈ S n[S 2], if there exists an element θ ∈ S n[S 2] such that ξ 2 = θξ 1θ −1,

then |ΦO( ξ 1)| = |ΦO( ξ 2)| and |ΦO( ξ )| = |ΦO( ξα)|. Notice that |ΦO( ξ )| has been gotten by

Lemma 8.6.1. Applying Lemma 8.6.1(1) and the cycle index Z (S n[S 2]), we get that

nO( Bn) =1

2 × 2nn!(

k |2n,k 2n

k 2nk

−1(2n

k − 1)!|J k | + φ(2n)|J 2n|)




=

k |2n,k 2n

k 2nk

−1(2n

k − 1)!

1

( 2nk

)!

∂2nk ( Z (S n[S 2]))

∂s2nk

k

|sk =0

+φ(2n)∂( Z (S n[S 2]))

∂s2n |s2n=0

Now we consider maps on non-orientable surfaces underlying graph Bn. Similar to

the discussion of Theorem 8.6.1, we get the following enumeration result for the maps on

non-orientable surfaces.

Theorem 8.6.2 For an integer n ≥ 1 , the number n N ( Bn) of maps on non-orientable

surfaces underlying graph Bn is

n N ( Bn) =(2n − 1)!

n!+

k

|2n,3

≤k <2n

(2k )2nk

−1(2n

k − 1)!

∂2nk ( Z (S n[S 2]))

∂s2nk

k

|sk =0

+1

2nn!(s≥1

n!

(n − 2s)!s!+ 4n(n − 1)!(

∂n( Z (S n[S 2]))

∂sn2

|s2=0 − ⌊n

2⌋)).

Proof Similar to the proof of Theorem 8.6.1, applying formula (1.3.3) in Corollary

8.3.1 and Lemma 8.6.1(2), we get that

n L( Bn) =(2n − 1)!

n!+ φ(2n)

∂n( Z (S n[S 2]))

∂sn2n

|s2n=0

+1

2nn!(

s≥0

n!

(n

−2s)!s!

+ 4n(n − 1)!(∂n( Z (S n[S 2]))

∂sn2

|s2=0 − ⌊n

2⌋))

+

k |2n,3≤k <2n

(2k )2nk

−1(2n

k − 1)!

∂2nk ( Z (S n[S 2]))

∂s2nk

k

|sk =0.

Notice that nO( Bn) + n N ( Bn) = n L( Bn). Applying Theorem 8.6.1, we find that

n N ( Bn) =(2n − 1)!

n!+

k |2n,3≤k <2n

(2k )2nk

−1(2n

k − 1)!

∂2nk ( Z (S n[S 2]))

∂s2nk

k

|sk =0

+1

2nn!(

s≥1

n!

(n − 2s)!s!+ 4n(n − 1)!(

∂n( Z (S n[S 2]))

∂sn2

|s2=0 − ⌊n

2⌋)).



Z (S 1[S 2]) =s2

1 + s2

2

and

Z (S 2[S 2]) =s4

1 + 2s21 s2 + 3s2

2 + 2s4

8,




Whence, if n = 2, calculation shows that there are 1 map on the plane, 2 maps on the

projective plane, 1 map on the torus and 2 maps on the Klein bottle. All of those maps are

non-isomorphic and the same as gotten by Theorems 8.6.1 and 8.6.2 shown in Fig.8.6.1.

1

1

2 2

1

1

2 2

11

2

2

3

3

4

4

5 5

¹

¹

¹

¹

Fig.8.6.1

§8.7 REMARKS

8.7.1 The enumeration problem of maps was first introduced by Tutte on planar rooted

triangulation by solving a functional equation in 1962. After him, more and more papers

and enumeration result on rooted maps on surfaces published. For surveying such an

enumeration, the readers are refereed to references [Liu2]-[Liu4] for details.

8.7.2 The enumeration of rooted maps on surfaces is canonically by an analytic approach.

Usually, this approach for enumeration of rooted maps applies four steps as follows:

STEP 1. Decompose the set of rooted maps M considered;

STEP 2. Define the enumeration function f M on maps by parameters, such as those of order n( M ), size m( M ), valency of rooted vertex or rooted face, · · · of maps, for example,

f M =

M ∈M xn( M ), f M =

M ∈M

xm( M ), f M =

M ∈M xn( M ) ym( M ) and f M =

M ∈M

xn( M ) ym( M ) zl( M )

are four enumeration functions respectively by order n( M ), size m( M ) and valency of

rooted vertex l( M ) of map and then establish equations satisfied by f M.



Sec.8.7 Remarks 305

STEP 3. Find properly parametric expression for variables x, y, z, · · ·.STEP 4. Applying the Lagrange inversion, i.e., if x = t φ( x) with φ(0) 0, then

f ( x)=

f (0)+i≥1

t i

i!

d i−1

dxi−1 φi d f

dx | x=0

solves the equations for enumeration.

The importance of Theorems 8.1.7 and 8.1.8 is that they clarify the essence of the

enumeration of rooted maps on surfaces, i.e., a calculation of the summation

G∈G

2ε(G)

v∈V (G)( ρ(v) − 1)!

|Aut 12

G| orG∈G

2 β(G)+1ε(G)

v∈V (G)( ρ(v) − 1)!

|Aut 12

G|

where G denotes a graph family. For example, we know that the number of rooted tree of size n is

(2n)!

n!(n + 1)!. Whence,

T ∈T (n)

d ∈ D(T )

(d − 1)!

|AutT| =(2n − 1)!

n!(n + 1)!,

where T and D(T ) denote sets of non-isomorphic trees of size n and the valency sequence

of a tree T ∈ T , respectively.

Similarly, Theorem 8.2.1 implies the enumeration of rooted maps on a surface S of

genus i is in fact a calculation of the summationG∈G(S )

2ε(G)gi(G)

|Aut 12

G| ,

where G(S ) denotes a graph family embeddable on S . For example, We know that there

are2(2n − 1)!(2n + 1)!

(n + 2)!(n + 1)!!n!(n − 1)!

planar cubic hamiltonian rooted maps. Whence,

G∈C H

2ε(G)g0(G)

|AutG| =2(2n − 1)!(2n + 1)!

(n + 2)!(n + 1)!!n!(n − 1)!,

where C H denotes the family of hamiltonian cubic.

8.7.3 By applying Burnside lemma, Biggs and White suggested a scheme for enumerat-

ing non-equivalent embeddings of a graph G on surfaces, i.e., orbits under the action of




AutG on all embeddings of G in [BiW1]. Such an action is in fact orientation-preserving.

Theorem 8.3.2 is a generalization of their result by considering the action of Aut 12

G × αon all embeddings of G on surfaces. This scheme enables one to find non-isomorphic

maps on surfaces underlying a graph. Indeed, complete maps, semi-regular maps andone-vertex maps are enumerated in Sections 8.4-8.6. Certainly, there are more maps on

surfaces needed to enumerated, such as those of maps included in problems following.

Problem 8.7.1 Enumerate maps on surfaces underlying a vertex-transitive, an edge-

transitive or a regular graph, particularly, a Cayley graph Cay(Γ : S ).

Problem 8.7.2 Enumeration maps on surfaces underlying a graph G with known Aut 12

G,

such as those of C n × P2 and C m × C n × C l for integers n, m, l ≥ 1.

Problem 8.7.3 Enumerate a typical maps underlying a graph, for example, regular mapsor Cayley maps.

The enumeration of maps on surfaces underlying a graph also brings about problems

following on graphs.

Problem 8.7.4 Find a graph family G on a surface S such that the number of non-

isomorphic maps underlying graph in G is summable.

Problem 8.7.5 For a surface S and an integer n ≥ 2 , determine the family Gn(S ) embed-

dable on S with |Aut 12 | = n for ∀G ∈ Gn(S ).



CHAPTER 9.

Isometries on Smarandache Geometry

We have known that classical geometry includes those of Euclid geometry,Lobachevshy-Bolyai-Gauss geometry and Riemann geometry. Each of the

later two is proposed by denial the 5th postulate for parallel lines in Euclid

postulates on geometry. For generalizing classical geometry, a new geometry,

called Smarandache geometry was proposed by Smarandache in 1969, which

may enables these three geometries to be united in the same space altogether

such that it can be either partially Euclidean and partially non-Euclidean, or

non-Euclidean. Such a geometry is really a hybridization of these geome-

tries. It is important for destroying the law that all points are equal in status

and introducing contradictory laws in a same geometrical space. For an in-

troduction to such geometry, we formally define Smarandache geometry, par-

ticularly, those of mixed geometries in Section 9.1, and classify s-manifolds,

a kind of Smarandache 2-manifolds by applying planar maps in Section 2.

After then, Sections 3 and 4 concentrate on the isometries on finite or infi-

nite pseudo-Euclidean spaces (Rn, µ) by verifying the action of isometries of

Rn on (Rn, µ) for n ≥ 2. Certainly, all isometries on finite pseudo-Euclidean

spaces (Rn, µ) are automorphisms of (Rn, µ), and can be characterized combi-

natorially by that of maps on surfaces if n = 2 or embedded graphs in Rn if

n ≥ 3.



308 Chap.9 Isometries on Smarandache Geometry

§9.1 SMARANDACHE GEOMETRY

9.1.1 Geometrical Axiom. As we known, the Euclidean geometrical axiom system

consists of five axioms following:

(E1) There is a straight line between any two points.

(E2) A finite straight line can produce a infinite straight line continuously.

(E3) Any point and a distance can describe a circle.

(E4) All right angles are equal to one another.

(E5) If a straight line falling on two straight lines make the interior angles on the

same side less than two right angles, then the two straight lines, if produced indefinitely,

meet on that side on which are the angles less than the two right angles.

The last axiom (E5) is usually replaced by:

(E5’) For a given line and a point exterior this line, there is one line parallel to this

line.

Then a hyperbolic geometry is replaced axiom (E5) by (L5) following

(L5) There are infinitely many lines parallel to a given line passing through an

exterior point,

and an elliptic geometry is replaced axiom (E5) by ( R5) following:

There are no parallel to a given line passing through an exterior point.

9.1.2 Smarandache Geometry. These non-Euclidean geometries constructed in the

previous subsection implies that one can find more non-Euclidean geometries replacing

Euclidean axioms by non-Euclidean axioms. In fact, a Smarandache geometry is such a

geometry by denied some axioms (E1)-(E5) following.

Definition 9.1.1 A rule R ∈ R in a mathematical system (Σ; R) is said to be Smaran-

dachely denied if it behaves in at least two di ff erent ways within the same set Σ , i.e.,validated and invalided, or only invalided but in multiple distinct ways.

Definition 9.1.2 A Smarandache geometry is such a geometry in which there are at

least one Smarandachely denied ruler and a Smarandache manifold ( M ; A) is an n-

dimensional manifold M that support a Smarandache geometry by Smarandachely denied

axioms in A.



Sec.9.1 Smarandache Geometry 309

In a Smarandache geometry, points, lines, planes, spaces, triangles, · · · are called

respectively s-points, s-lines, s-planes, s-spaces, s-triangles, · · · in order to distinguish

them from that in classical geometry.

Example 9.1.1 Let us consider a Euclidean plane R2 and three non-collinear points A, B

and C . Define s-points as all usual Euclidean points on R2 and s-lines any Euclidean line

that passes through one and only one of points A, B and C . Then such a geometry is a

Smarandache geometry by the following observations.

Observation 1. The axiom (E1) that through any two distinct points there exist

one line passing through them is now replaced by: one s-line and no s-line. Notice that

through any two distinct s-points D, E collinear with one of A, B and C , there is one

s-line passing through them and through any two distinct s-points F , G lying on AB ornon-collinear with one of A, B and C , there is no s-line passing through them such as

those shown in Fig.9.1.1(a).

Observation 2. The axiom (E5) that through a point exterior to a given line there is

only one parallel passing through it is now replaced by two statements: one parallel and

no parallel. Let L be an s-line passes through C and is parallel in the Euclidean sense to

AB. Notice that through any s-point not lying on AB there is one s-line parallel to L and

through any other s-point lying on AB there is no s-lines parallel to L such as those shown

in Fig.9.1.1(b).

L

l1

l2

D

BA

C

E

(b)(a)

D C E

A BF G

l1

Fig.9.1.1

9.1.3 Mixed Geometry. In references [Sma1]-[Sma2], Smarandache introduced a

few mixed geometries, such as those of the paradoxist geometry, the non-geometry, the

counter-projective geometry and the anti-geometry by contradicts axioms ( E 1) − ( E 5)

in a Euclid geometry following. All of these geometries are examples of Smarandache

geometry.




Paradoxist Geometry. In this geometry, its axioms consist of ( E 1) − ( E 4) and one of the

following:

(1) There are at least a straight line and a point exterior to it in this space for which

any line that passes through the point intersect the initial line.


only one line passes through the point and does not intersect the initial line.


only a finite number of lines l1, l2, · · · , lk , k ≥ 2 pass through the point and do not intersect

the initial line.


an infinite number of lines pass through the point (but not all of them) and do not intersect

the initial line.


any line that passes through the point and does not intersect the initial line.

Non-Geometry. The non-geometry is a geometry by denial some axioms of ( E 1)− ( E 5),

such as those of the following:

( E 1−) It is not always possible to draw a line from an arbitrary point to another

arbitrary point.

( E 2−) It is not always possible to extend by continuity a finite line to an infinite line.

( E 3−) It is not always possible to draw a circle from an arbitrary point and of an

arbitrary interval.

( E 4−) Not all the right angles are congruent.

( E 5−) If a line cutting two other lines forms the interior angles of the same side of it

strictly less than two right angle, then not always the two lines extended towards infinite

cut each other in the side where the angles are strictly less than two right angle.

Counter-Projective Geometry. Denoted by P the point set, L the line set and R a relation

included in P× L. A counter-projective geometry is a geometry with these counter-axioms(C 1) − (C 3) following:

(C 1) There exist either at least two lines, or no line, that contains two given distinct

points.

(C 2) Let p1, p2, p3 be three non-collinear points and q1, q2 two distinct points. Sup-

pose that p1.q1, p3 and p2, q2, p3 are collinear triples. Then the line containing p1, p2



Sec.9.2 Classifying Iseri’s Manifolds 311

and the line containing q1, q2 do not intersect.

(C 3) Every line contains at most two distinct points.

Anti-Geometry. A geometry by denial some axioms of the Hilbert’s 21 axioms of Eu-

clidean geometry.

§9.2 CLASSIFYING ISERI’S MANIFOLDS

9.2.1 Iseri’s Manifold. The idea of Iseri’s manifolds was based on a paper [Wee1] and

credited to W.Thurston. A more general idea can be found in [PoS1]. Such a manifold is

combinatorially defined in [Ise1] as follows:

An Iseri’s manifold is any collection C(T , n) of these equilateral triangular disks

T i, 1 ≤ i ≤ n satisfying the following conditions:

(1) Each edge e is the identification of at most two edges ei, e j in two distinct trian-

gular disks T i, T j, 1 ≤ i, j ≤ n and i j;

(2) Each vertex v is the identification of one vertex in each of five, six or seven

distinct triangular disks.

The vertices of an Iseri’s manifold are classified by the number of the disks around

them. A vertex around five, six or seven triangular disks is called an elliptic vertex, a Euclid vertex or a hyperbolic vertex, respectively.

An Iseri’s manifold is called closed if the number of triangular disks is finite and

each edge is shared by exactly two triangular disks, each vertex is completely around

by triangular disks. It is obvious that a closed Iseri’s manifold is a surface and its Euler

characteristic can be defined by Theorem 4.2.6.

Two Iseri’s manifolds C1(T , n) and C2(T , n) are called to be isomorphic if there is an

1 − 1 mapping τ : C1(T , n) → C2(T , n) such that for ∀T 1, T 2 ∈ C1(T , n), τ(T 1

T 2) =

τ(T 1

) τ(T 2

). If C1

(T , n) =C1

(T , n) =C

(T , n), τ is called an automorphism of Iseri’s

manifold C(T , n). All automorphisms of an Iseri’s manifold form a group under the com-

position operation, called the automorphism group of C(T , n) and denoted by AutC(T , n).

9.2.2 A Model of Closed Iseri’s Manifold. For a closed Iseri’s manifold C(T , n), we

can define a map M by V ( M ) = the vertices in C(T , n), E ( M ) = the edges in C(T , n)and F ( M ) = T , T ∈ C(T , n). Then M is a triangular map with vertex valency ∈ 5, 6, 7.




On the other hand, if M is a triangular map on surface with vertex valency∈ 5, 6, 7, we

can define an Iseri’s manifold C(T , φ( M )) by

C(T , φ( M )) =

f | f

∈F ( M )

.

Then C(T , φ( M )) is an Iseri’s manifold. Consequently, we get a result following.

Theorem 9.2.1 Let C(T , n) , M(T , n) and M∗(T , n) be the set of Iseri’s manifolds with n

triangular disks, triangular maps with n faces and vertex valency ∈ 5, 6, 7 and cubic

maps of order n with face valency ∈ 5, 6, 7. Then

(1) There is a bijection between M(T , n) and C(T , n);

(2) There is also a bijection between M∗(T , n) and

C(T , n).

According to Theorem 9.2.1, we get the following result for the automorphisms of

an Iseri’s manifold following.

Theorem 9.2.2 Let C(T , n) be a closed s-manifold with negative Euler characteristic.

Then |AutC(T , n)| ≤ 6n and

|AutC(T , n)| ≤ −21 χ(C(T , n)),

with equality hold only if C(T , n) is hyperbolic, where χ(C(T , n)) denotes the genus of

C(T , n).

Proof The inequality |AutC(T , n)| ≤ 6n is known by the Corollary 6.4.1. Similar to

the proof of Theorem 6.4.2, we know that

ε(C(T , n)) =− χ(C(T , n))

13− 2

k

,

where k =1

ν(C(T , n))

i≥1

iνi ≤ 7 and with the equality holds only if k = 7, i.e., C(T , n) is

hyperbolic.

9.2.3 Classifying Closed Iseri’s Manifolds. According to Theorem 9.2.1, we can clas-

sify closedIseri’s manifolds by that of triangular maps with valency in 5, 6, 7 as follows:

Classical Type:

(1) ∆1 = 5 − regular triangular maps (elliptic);

(2) ∆2 = 6 − regular triangular maps(euclid );






Proof If M is a k -regular triangulation, we get that 2ε( M ) = 3φ( M ) = k ν( M ).

Whence, we have

ε( M ) =3φ( M )

2and ν( M ) =

3ε( M )

k .

By the Euler-Poincare formula, we know that

χ( M ) = ν( M ) − ε( M ) + φ( M ) = (3

k − 1

2)φ( M ).

If M is elliptic, then k = 5. Whence, χ( M ) =φ( M )

10> 0. Therefore, if M is orientable,

then χ( M ) = 2, Whence, φ( M ) = 20, ν( M ) = 12 and ε( M ) = 30, which is just the

map O20. If M is non-orientable, then χ( M ) = 1, Whence, φ( M ) = 10, ν( M ) = 6 and

ε( M ) = 15, which is the map P10.

If M is Euclidean, then k = 6. Thus χ( M ) = 0, i.e., M is a 6-regular triangulation T i

or K j for some integer i or j on the torus or Klein bottle, which is infinite.

If M is hyperbolic, then k = 7. Whence, χ( M ) < 0. M is a 7-regular triangulation,

i.e., the Hurwitz map. According to the results in [Sur1], there are infinite Hurwitz maps

on surfaces. This completes the proof.

For these Smarandache Types, the situation is complex. But we can also obtain the

enumeration results for each of the types ∆4 - ∆7. First, we prove a condition for the

numbers of vertex valency 5 with that of 7.

Lemma 9.2.1 Let C(T , n) be an Iseri’s manifold. Then

v7 ≥ v5 + 2

if χ(C(T , n)) ≤ −1 and

v7 ≤ v5 − 2

if χ(C(T , n)) ≥ 1 , where vi denotes the number of vertices of valency i in C(T , n).

Proof Notice that we have know

ε(C(T , n)) = − χ(C(T , n))13− 2

k

,

where k is the average valency of vertices in C(T , n). Since

k =5v5 + 6v6 + 7v7

v5 + v6 + v7

and ε(C(T , n)) ≥ 3. Consequently, we get that




(1) If χ(C(T , n)) ≤ −1, then

1

3− 2v5 + 2v6 + 2v7

5v5 + 6v6 + 7v7

> 0,

i.e., v7 ≥ v5 + 1. Now if v7 = v5 + 1, then

5v5 + 6v6 + 7v7 = 12v5 + 6v6 + 7 ≡ 1(mod 2).

Contradicts to the fact that v∈V (G)

ρG(v) = 2ε(G) ≡ 0(mod 2)

for a graph G. Whence there must be

v7 ≥ v5 + 2.

(2) If χ(C(T , n)) ≥ 1, then

1

3− 2v5 + 2v6 + 2v7

5v5 + 6v6 + 7v7

< 0,

i.e., v7 ≤ v5 − 1. Now if v7 = v5 − 1, then

5v5 + 6v6 + 7v7 = 12v5 + 6v6

−7

≡1(mod 2).

Also contradicts to the fact thatv∈V (G)

ρG(v) = 2ε(G) ≡ 0(mod 2)

for a graph G. Whence, there must be

v7 ≤ v5 −2.

Corollary 9.2.1 There are no Iseri’s manifolds C(T , n) such that

|v7 − v5| ≤ 1,

where vi denotes the number of vertices of valency i in C(T , n).

Define an operator Θ : M → M ∗ on a triangulation M of a surface by




Choose each midpoint on each edge in M and connect the midpoint in each triangle

as shown in Fig.9.2.3. Then the resultant M ∗ is a triangulation of the same surface and

the valency of each new vertex is 6.

Θ

M M ∗

Fig. 9.2.3

Then we get the following result.

Theorem 9.2.4 For these Smarandache Types ∆4-∆7 , there are

(1) |∆5| ≥ 2;

(2) Each of |∆4|, |∆6| and |∆7| is infinite.

Proof For M ∈ ∆4, let k be the average valency of vertices in M . Since

k =5v5 + 6v6

v5 + v6

< 6 and ε( M ) =− χ( M )

1

3− 2

k

,

we have that χ( M ) ≥ 1. Calculation shows that v5 = 6 if χ( M ) = 1 and v5 = 12 if

χ( M ) = 2. We can construct a triangulation with vertex valency 5, 6 on the plane and the

projective plane in Fig.9.2.4.

1

7

234

5

6

7

1

2 3 4 5

6

(a) (b)

¹

Fig.9.2.4

Now let M be a map in Fig.9.2.4. Then M Θ is also a triangulation of the same surface




with vertex valency 5, 6 and M Θ M . Whence, |∆4| is infinite.

For M ∈ ∆5, by the Lemma 9.2.1, we know that v7 ≤ v5 − 2 if χ( M ) ≥ 1 and

v7 ≥ v5 + 2 if χ( M ) ≤ −1. We construct a triangulation on the plane and projective plane

in Fig.9.2.5.

1

123

4

56

2

7

34

5

6

7

¹

Fig.9.2.5

For M ∈ ∆6, we know that k =6v6 + 7v7

v6 + v7

> 6. Whence, χ( M ) ≤ −1. Since

3φ( M ) = 6v6 + 7v7 = 2ε( M ), we get that

v6 + v7 − 6v6 + 7v7

2+

6v6 + 7v7

3= χ( M ).

Therefore, we have v7 = − χ( M ). Notice that there are infinite Hurwitz maps M on sur-

faces. Then the resultant triangular map M ∗ is a triangulation with vertex valency 6, 7 and

M ∗ M . Thus |∆6| is infinite.

For M

∈∆7, we construct a triangulation with vertex valency 5, 6, 7 in Fig.9.2.6.

12

3

456

1

2

3

4

7

5 6

7¹

Fig.9.2.6

Let M be one of the maps in Fig.9.2.6. Then the action of Θ on M results infinite

triangulations of valency 5, 6 or 7. This completes the proof.

For the set ∆5, we have the following conjecture.

Conjecture 9.2.1 The number |∆5| is infinite.




§9.3 ISOMETRIES OF SMARANDACHE 2-MANIFOLDS

9.3.1 Smarandachely Automorphism. Let ( M ; A) be a Smarandache manifold. By

definition a Smarandachely denied axiom A∈ A

can be considered as an action of A

on subsets S ⊂ M , denoted by S A. Now let ( M 1; A1) and ( M 2; A2) be two Smarandache

manifolds, where A1, A2 are the Smarandachely denied axioms on manifolds M 1 and M 2,

respectively. They are said to be isomorphic if there is 1 − 1 mappings τ : M 1 → M 2 and

σ : A1 → A2 such that τ(S A) = τ(S )σ(A) for ∀S ⊂ M 1 and A ∈ A1. Such a pair (τ, σ) is

called an isomorphism between ( M 1; A1) and ( M 2; A2). Particularly, if M 1 = M 2 = M and

A1 = A2 = A, such an isomorphism (τ, σ) is called a Smarandachely automorphism of

( M , A). Clearly, all such automorphisms of ( M , A) form an group under the composition

operation on τ for a given σ. Denoted by Aut( M ,

A).

9.3.2 Isometry on R2. Let X be a set and ρ : X × X → R a metric on X , i.e.,

(1) ρ( x, y) ≥ 0 for x, y ∈ X , and with equality hold if and only if x = y;

(2) ρ( x, y) = ρ( y, x) for x, y ∈ X ;

(3) ρ( x, y) + ρ( y, z) ≥ ρ( x, z) for x, y, z ∈ X .

A set X with such a metric ρ is called a metric space, denoted by ( X , ρ).

Example 9.3.1 Let R2 = ( x, y) | x, y ∈ R . Define

ρ(x1, x2) = ( x1 − x2)2 + ( y1 − y2)2

for x1 = ( x1, y1), x2 = ( x2, y2) ∈ R2. Then such a ρ is a metric on R2. We verify conditions

(1)-(3) in the following.

Clearly, conditions (1) and (2) are consequence of x2 = 0 ⇒ x = 0 and x2 = (− x)2

for x ∈ R. Now let ( x1, y1), ( x2, y2) and ( x3, y3) be three points on R2 with

( x2, y2) = ( x1 + a1, y1 + b1)

( x3, y3) = ( x1 + a1 + a2, y1 + b1 + b2)

Then the condition (3) implies thata2

1 + b21 +

a2

2 + b22≥

(a1 + a2)2 + (b1 + b2)2,

which can be verified to be hold immediately.



Sec.9.3 Isometries on Smarndache 2-Manifolds 319

An isometry of a metric space ( X , ρ) is a bijective mapping φ : X → X that preserves

distance, i.e., ρ(φ(x), φ(y)) = ρ(x, y). Denote by Isom( X , ρ) the set of all isometries of

( X , ρ). Then we know the following.

Theorem 9.3.1 Isom( X , ρ) is a group under the composition operation of mapping.

Proof Clearly, 1 X ∈ Isom( X ) and if φ ∈ Isom( X ), then φ−1 ∈ Isom( X ). Furthermore,

if φ1, φ2 ∈ Isom( X ), by definition we know that

ρ(φ1φ2(x), φ1φ2(y)) = ρ(φ2(x), φ2(y)) = ρ(x, y).

Whence, φ1φ2 is also an isometry, i.e., φ1φ2 ∈ Isom( X ). So Isom( X , ρ) is a group.

Let ∆, ∆′ be two triangles on R2. They are said to be congruent if we can label their

vertices, for instance ∆ = ABC and ∆′ = A′ B′C ′ such that

| AB| = | A′ B′|, | BC | = | B′C ′|, |CA| = |C ′ A′|,∠CA B = ∠C ′ A′ B′, ∠ ABC = ∠ A′ B′C ′, ∠ BCA = ∠ B′C ′ A′.

Theorem 9.3.2 Let φ be an isometry on R2. Then φ maps a triangle to its a congruent

triangle, preserves angles and maps lines to lines.

Proof Let ∆ be a triangle with vertex labels A, B and C on R2. Then φ(∆) is congruent

with ∆ by the definition of isometry.

Notice that an angle ∠ < π and an angle ∠ > π can be realized respectively as an

angle ∠CA B, or an exterior angle of a triangle ABC . We have known that φ( ABC ) is

congruent with ABC . Consequently, ∠φ(C )φ( A)φ( B) = ∠CAB, i.e., φ preserves angles in

R2. If ∠ = π, this result follows the law of trichotomy.

For a line L in R2, let B, C be two distinct points on L, and let L′ be the line through

points B′ = φ( B) and C ′ = φ(C ). Then for any point A ∈ R2, it follows that

φ( A) φ( L) ⇔ A L ⇔ 0 ≤ ∠CAB < π

⇔ 0 < ∠C ′φ( A) B′ < π ⇔ φ( A) L′.

Therefore, φ( L) = L′.

The behavior of an isometry is completely determined by its action on three non-

collinear points shown in the next result.




Theorem 9.3.3 An isometry of R2 is determined by its action on three non-collinear

points.

Proof Let A, B, C be three non-collinear points on R2 and let φ1, φ2

∈Isom(R2)

have the same action on A, B, C . Thus

φ1( A) = φ2( A), φ1( B) = φ2( B), φ1(C ) = φ2(C ).

i.e.,,

φ−12 φ1( A) = A, φ−1

2 φ1( B) = B, φ−12 φ1(C ) = C .

Whence, we must show that if there exists ϕ ∈ Isom(R2) such that ϕ( A) = A, ϕ( B) =

B, ϕ(C ) = C , then ϕ(P) = P for each point P ∈ R2.

In fact, since ϕ preserves distance and ϕ( A) = A, it follows that P and ϕ(P) are

equidistant from A. Thus ϕ(P) lies on the circle C 1 centered at A with radius | AP|. Sim-

ilarly, ϕ(P) also lies on the circle C 2 centered at B with radius | BP|. Whence, ϕ(P) ∈C 1 ∩ C 2.

Because C 1 and C 2 are not concentric, they intersect in at most two points, such as

those shown in Fig.9.3.1 following.

A B

ϕ(P)

P

C 1C 2

C

L

Fig.9.3.1

Notice that P lies on both of C 1 and C 2. Thus C 1 ∩ C 2 ∅. Therefore, |C 1 ∩ C 2| = 1

or 2. If |C 1 ∩ C 2| = 1, then ϕ(P) = P. If |C 1 ∩ C 2| = 2, let L be the line through A, B,

which is the perpendicular bisector of ϕ(P) and P, such as those shown in Fig.9.3.1. By

assumption, C L, we get that |CP| |C ϕ(P)|. Contradicts to the fact that P, ϕ(P) are

equidistant from C . Whence |C 1 ∩ C 2| = 1 and we get the conclusion.




There are three types of isometries on R2 listed in the following.

Translation T. A translation T is a mapping that moves every point of R2 through

a constant distance in a fixed direction, i.e.,

T a,b : R2 → R2, ( x1, y1) → ( x1 + a, y1 + b),

where (a, b) is a constant vector. Call the direction of (a, b) the axis of T and denoted by

T = T a,b.

Rotation Rθ . A rotation R is a mapping that moves every point of R2 through a

fixed angle about a fixed point, called the center . By taking the center O to be the origin

of polar coordinates (r , θ ), a rotation Rθ : R2 → R2 is

R : (r , θ ) → (r , θ + ),

where is a constant angle, ∈ R (mod2π). Denoted by R = Rθ .

Reflection F. A reflection F is a mapping that moves every point of R2 to its mirror-

image in a fixed line. That line L is called the axis of F , denoted by F = F ( L). Thus for a

point P in R2, if P ∈ L, then F (P) = P, and if P L, then F (P) is the unique point in R2

such that L is the perpendicular bisector of P and F (P).

Theorem 9.3.4 For a chosen line L and a fixed point O

∈L in R2 , any element ϕ

∈Isom(R2) can written uniquely in the form

ϕ = T RF ǫ ,

where F denotes the reflection in L, ǫ = 0 or 1 , R is the rotation centered at O, T ∈ T , and

the subgroup of orientation-preserving isometries of R2 consists of those ϕ with ǫ = 0.

Proof Let T be the translation transferring O to ϕ(O). Clearly, T −1ϕ(O) = O. Now

let P ∈ L be a point with P O. By definition,

0 < ρ(O, P) = ρ(T −1ϕ(O), T −1ϕ(P)) = ρ(O, T −1ϕ(P)),

there exists a rotation R centered at O transferring P to T −1ϕ(P). Thus R−1T −1ϕ fixes both

points O and P.

Finally, let Q L be a point. Then points Q and R−1T −1ϕ(Q) are equidistant both

from points O and P. Similar to the proof of Theorem 9.3.3, we know that points Q and




R−1T −1ϕ(Q) are either equal or mirror-images in L. Choose ǫ = 0 if Q = R−1T −1ϕ(Q) and

ǫ = 1 if Q R−1T −1ϕ(Q). Then the isometry F ǫ R−1T −1ϕ fixes non-collinear points O, P

and Q. According to Theorem 9.3.3, there must be

F ǫ R−1T −1ϕ = 1R2 .

Thus

ϕ = T RF ǫ .

For the uniqueness of the form, assume that

T RF ǫ = T ′ R′F δ,

where ǫ, δ ∈ 0, 1, T , T ′ ∈ T and R, R′ ∈ RO. Clearly, ǫ = δ by previous argument.Cancelling F if necessary, we get that T R = T ′ R′. But then (T ′)−1T = R′ R−1 belongs

to RO ∩ T, i.e., a translation fixes point O. Whence, it is the identity mapping 1r2 . Thus

T = T ′ and R = R′.

Notice that T , R are orientation-preserving but F is orientation-reversing. It follows

that T RF ǫ is orientation-preserving or orientation-reversing according to ǫ = 0 or 1. This


9.3.3 Finitely Smarandache 2-Manifold. A point P on a Euclidean plane R2 is in fact

associated with a real number π. Generally, we consider a function µ : R2 → [0, 2π) and

classify points on R2 into three classes following:

Elliptic Type. All points P ∈ R2 with µ(P) < π.

Euclidean Type. All points Q ∈ R2 with µ(P) = π.

Hyperbolic Type. All points U ∈ R2 with µ(P) > π.

Such a Euclidean plane R2 with elliptic or hyperbolic points is called a Smarandache

plane, denoted by (R2, µ) and these elliptic or hyperbolic points are called non-Euclidean

points. A finitely Smarandache plane is such a Smarandache plane with finite non-Euclidean points.

Let L be an s-line in a Smarandache plane (R2, µ) with non-Euclisedn points A1, A2,

· · · , An for an integer n ≥ 0. Its curvature R( L) is defined by

R( L) =

ni=1

(π − µ( Ai)).




An s-line L is called Euclidean or non-Euclidean if R( L) = ±2π or ±2π. The following

result characterizes s-lines on (R2, µ).

Theorem 9.3.5 An s-line without self-intersections is closed if and only if it is Euclidean.

Proof Let (R2, µ) be a Smarandache plane and let L be a closed s-line without self-

intersections on (R2, µ) with vertices A1, A2, · · · , An. From the Euclid geometry on plane,

we know that the angle sum of an n-polygon is (n − 2)π. Whence, the curvature R( L) of

s-line L is ±2π by definition, i.e., L is Euclidean.

Now if an s-line L is Euclidean, then R( L) = ±2π by definition. Thus there exist

non-Euclidean points B1, B2, · · · , Bn such that

n

i=1

(π − µ( Bi)) = ±2π.

Whence, L is nothing but an n-polygon with vertices B1, B2, · · · , Bn on R2. Therefore, L

is closed without self-intersection.

Furthermore, we find conditions for an s-line to be that of regular polygon on R2

following.

Corollary 9.3.1 An s-line without self-intersection passing through non-Euclidean points

A1, A2, · · · , An is a regular polygon if and only if all points A1, A2, · · · , An are elliptic with

µ( Ai) = 1 −2

n π

or all A1, A2, · · · , An are hyperbolic with

µ( Ai) =

1 +

2

n

π

for integers 1 ≤ i ≤ n.

Proof If an s-line L without self-intersection passing through non-Euclidean points

A1, A2, · · · , An is a regular polygon, then all points A1, A2, · · · , An must be elliptic (hyper-

bolic) and calculation easily shows that

µ( Ai) =1 − 2

n

π or µ( Ai) =

1 +

2

n

π

for integers 1 ≤ i ≤ n by Theorem 9.3.5. On the other hand, if L is an s-line passing

through elliptic (hyperbolic) points A1, A2, · · · , An with

µ( Ai) =

1 − 2

n

π or µ( Ai) =

1 +

2

n

π




for integers 1 ≤ i ≤ n, then it is closed by Theorem 9.3.5. Clearly, L is a regular polygon

with vertices A1, A2, · · · , An.

Let ρ be the metric on R2 defined in Example 9.3.1. An isometry on a Smarandache

plane (R2, µ) is such an isometry τ : R2 → R2 with µ(τ( x)) = µ( x) for x ∈ R2. Clearly,

all isometries on (R2, µ) also form a group under the composition operation, denoted by

Isom(R2, µ). Corollary 9.3.1 enables one to determine isometries of finitely Smarandache

planes following.

Theorem 9.3.6 Let (R2, µ) be a finitely Smarandache plane. Then any isometry T of

(R2, µ) is generated by a rotation R and a reflection F on R2 , i.e., T = RF ǫ with ǫ = 0, 1.

Proof Let T be an isometry on a finitely Smarandache plane (R2, µ). Then for a

point A on (R

2

, µ), the type of A and T ( A) must be the same with µ(T ( A)) = µ( A) bydefinition. Whence, if there is constant vector (a, b) ∈ R2 such that T a,b : (R2, µ) →(R2, µ) determined by

( x, y) → ( x + a, y + b)

is an isometry and A a non-Euclidean point in (R2, µ), then there are infinite non-Euclidean

points A, T a,b( A), T 2a,b

( A), · · · , T na,b

( A), · · · , for integers n ≥ 1, contradicts the assumption

that (R2, µ) is finitely Smarandache. Thus T can be only generated by a rotation and a

refection. Thus T = RF ǫ . Conversely, we are easily constructing a rotation R and a

reflection F on (R2

, µ). For example, a rotation R : θ → θ + π/2 centered at O anda reflection F in line L on a finitely Smarandache plane (R2, µ) is shown in Fig.9.3.2

(a) and (b) in which the labeling number on a point P is µ(P) if µ(P) π. Otherwise,

µ(P) = π if there are no a label for p ∈ R2.

¹

π

2O

π

2

π2

π

2

π2

(a)

O

¹

π

2

π2

π

2

π2

L

(b)

Fig.9.3.2




The classification on finitely Smarandache planes is the following result.

Theorem 9.3.7 Let k |n or k |(n − 1) and 0 < d 1 < d 2 < · · · d k an integer sequence. Then

there exist one and only one finitely Smarandache plane (R2, µ) with n non-Euclidean

points A1, A2, · · · , An such that

Isom(R2, µ) ≃ D2k

and

ρ(O, Ai j) = d j, µ( Ai j

) =

1 − 2

k

, ( j − 1)k + 1 ≤ i j ≤ jk ; 1 ≤ j ≤ n

k

if k |n, or

ρ(O, Ai j) = d j, µ( Ai j

) = 1 −2

k , ( j

−1)k + 1

≤i j

≤jk ; 1

≤j

≤n − 1

k

with O = An if k |(n − 1).

Proof Choose =2π

k and a rotation R : (r , θ ) → (r , θ + ) centered at O.

Assume k |n. Let P1, P2, · · · , P nk

ben

k concentrically regular k -polygons at O with radius

d 1, d 2, · · · , d k . Place points A1, A2, · · · , Ak on vertices of P1, Ak 1 , Ak +2, · · · , A2k on vertices

of P2, · · ·, and An−k +1, An−k +2, · · · , An on vertices of P nk

, such as those shown in Fig.9.3.3.

O A1

A2 A3

Ak

Ak +1

Ak +2 Ak +3

A2k

An−k +1

An−k +2 An−k +3

An An−1

Fig.9.3.3

Then we are easily know that

Isom(R2, µ) ≃ D2k .

For the uniqueness, let P′1, P′

2, · · · , P′nk

ben

k concentrically regular k -polygons at O′

with radius d 1, d 2, · · · , d k and vertices A′1

, A′2

, · · · , A′n labeled likely that in Fig.9.3.3.




Choose T O′ ,O being a translation moving point O′ to O and R A′1

, A1a rotation centered at O

moving A′1 to A1. Transfer it first by T O′,O and then by R A′

1, A1

. Then each non-Euclidean

point A′i coincides with Ai for integers 1 ≤ i ≤ n, i.e., they are the same Smarandache

plane (R2

, µ).Similarly, we can get the result for the case of k |(n − 1) by putting O = An.

9.3.4 Smarandachely Map. Let S be a surface associated with µ : x → [0, 2π) for

each point x ∈ S , denoted by (S , µ). A point x ∈ S is called elliptic, Euclidean or

hyperbolic if it has a neighborhood U x homeomorphic to a 2-disk neighborhood of an

elliptic, Euclidean or a hyperbolic point in (R2, µ). Similarly, a line on (S , µ) is called an

s-line.

A map M = (X α,β, P) on (S , µ) is called Smarandachely if all of its vertices is

elliptic (hyperbolic). Notice that these pendent vertices is not important because it can

be always Euclidean or non-Euclidean. We concentrate our attention to non-separated

maps. Such maps always exist circuit-decompositions. The following result characterizes

Smarandachely maps.

Theorem 9.3.8 A non-separated planar map M is Smarandachely if and only if there

exist a directed circuit-decomposition

E 12

( M ) =

s

i=1

E (−→C i)

of M such that one of the linear systems of equationsv∈V (

−→C i)

(π − xv) = 2π, 1 ≤ i ≤ s

or v∈V (

−→C i)

(π − xv) = −2π, 1 ≤ i ≤ s

is solvable, where E 12

( M ) denotes the set of semi-arcs of M.

Proof If M is Smarandachely, then each vertex v ∈ V ( M ) is non-Euclidean, i.e.,

µ(v) π. Whence, there exists a directed circuit-decomposition

E 12

( M ) =

si=1

E (−→C i)




of semi-arcs in M such that each of them is an s-line in (R2, µ). Applying Theorem 9.3.5,

we know that

v∈V (−→C i)

(π − µ(v)) = 2π or

v∈V (−→C i)

(π − µ(v)) = −2π

for each circuit C i, 1 ≤ i ≤ s. Thus one of the linear systems of equationsv∈V (

−→C i)

(π − xv) = 2π, 1 ≤ i ≤ s or

v∈V (−→C i)

(π − xv) = −2π, 1 ≤ i ≤ s

is solvable.

Conversely, if one of the linear systems of equations

v∈V (

−→C i)

(π − xv) = 2π, 1 ≤ i ≤ s or v∈V (

−→C i)

(π − xv) = −2π, 1 ≤ i ≤ s

is solvable, define a mapping µ : R2 → [0, 4π) by

µ( x) =

xv if x = v ∈ V ( M ),

π if x v( M ).

Then M is a Smarandachely map on (R2, µ). This completes the proof.

In Fig.9.3.4, we present an example of a Smarandachely planar maps with µ defined

by numbers on vertices.

π

2

π

2

π

2

π

2

π

2

π

2

π

2

π

2

π

2

Fig.9.3.4

Let ω0 ∈ (0, π). An s-line L is called non-Euclidean of type ω0 if R( L) = ±2π ± ω0.

Similar to Theorem 9.3.8, we can get the following result.




Theorem 9.3.9 A non-separated map M is Smarandachely if and only if there exist a

directed circuit-decomposition

E 12 ( M ) =

s

i=1 E (−→C i)

of M into s-lines of type ω0 , ω0 ∈ (0, π) for integers 1 ≤ i ≤ s such that one of the linear

systems of equations v∈V (

−→C i)

(π − xv) = 2π − ω0, 1 ≤ i ≤ s,

v∈V (

−→C i)

(π − xv) = −2π − ω0, 1 ≤ i ≤ s,

v∈V (

−→C i)

(π − xv) = 2π + ω0, 1 ≤ i ≤ s,

v∈V (

−→C i)

(π − xv) = −2π + ω0, 1 ≤ i ≤ s

is solvable.

9.3.5 Infinitely Smarandache 2-Manifold. Notice that the function µ : R2 → [0, 2π)

is not continuous if there are only finitely non-Euclidean points in (R2, µ). We consider

a continuous function µ : R2 → [0, 2π) in this subsection, in which we meet infinitenon-Euclidean points.

¹

x

y

O

l1l2

r(s)

X

Y

φψ

δ

Fig.9.3.5

Let r : (a, b) → R2 be a plane curve C parametrized by arc length s, seeing Fig.9.3.5.

Notice that µ( x) is an angle variant from π of a Euclidean point to µ( x) of a non-Euclidean




x in finitely Smarandache plane. Consider points moves from X to Y on r(s). Then the

variant of angles from l1 to l2 is δ = φ − ψ. Thus µ( x) =d φ

ds

x

. Define the curvature R(C )

of curve C by

R(C ) = C

d φ

ds .

Then if C is a closed curve on R2 without self-intersection, we get that

R(C ) =

C

d φ

ds=

2πr 0

d φ

ds= φ|2πr − φ|0 = 2π.

Let r = ( x(s), y)(s)) be a plane curve in R2. Then

dx

ds = cos φ,

dy

d s = sin φ.

Consequently,

d 2 x

ds2= − sin φ

d φ

d s= −dy

d s

d φ

d s,

d 2 y

ds2= cos φ

d φ

ds=

d x

ds

d φ

ds.

Multiplying the first formula by −dy

ds, the second by

d x

dson both sides and plus them, we

get thatd φ

ds

=d x

ds

d 2 y

ds2

−

d 2 x

ds2

dy

d sby applying sin2 φ + cos2 φ = 1.

If r(t ) = ( x(t ), y(t )) is a plane curve C parametrized by t , where t maybe not the arc

length, since

s =

t 0

dx

dt

2

+

dy

dt

2

dt ,

we know that

ds

dt = dx

dt 2

+ dy

dt 2

,

d x

ds = dx

dt / d s

dt and

dy

ds = dy

dt / ds

dt .

Whence,

d φ

ds=

dx

dt

d 2 y

dt 2− d 2 x

dt 2dy

dt dx

dt

2

+

dy

dt

2 32

.




Consequently, we get the following result by definition.

Theorem 9.3.10 A curve C determined by r = ( x(t ), y)(t )) exists in a Smarandache plane

(R2, µ) if and only if the following di ff erential equation

dx

dt

d 2 y

dt 2− d 2 x

dt 2dy

dt dx

dt

2

+

dy

dt

2 32

= µ

is solvable.

Example 9.3.1 Let r(θ ) = (cos θ, sin θ ) (0 ≤ θ ≤ 2π) be a unit circle C on R2. Calculation

shows that

dxd θ

d 2

yd θ 2

− d 2

xd θ 2

dyd θ

= sin2 θ + cos2 θ = 1

and dx

dt

2

+

dy

dt

2 32

= sin2 θ + cos2 θ = 1.

Whence, the circle C exists in a Smarandache plane (R2, µ) if and only if µ( x, y) = 1 for

∀( x, y) ∈ C .

Example 9.3.2 Let r(t ) = (a(t − sin t ), a(1 − cos t )) (0 ≤ t ≤ 2π) be a spiral line on R2.

Calculation shows thatd φ

d s= − 1

4a sint

2

.

Whence, this spiral line exists in a Smarandache plane (R2, µ) if and only if

µ( x, y) = − 1

4a sint

2

for x = a(t − sin t ) and y = a(1 − cos t ).

Now we turn our attention to isometries of Smarandache plane (R2

, µ) with infinitelySmarandache points. These points in (R2, µ) can be classified into three classes, i.e.,

elliptic points V el , Euclidean points V eu and hyperbolic points V hy following:

V el = u ∈ (R2, µ) | µ(u) < π ,

V eu = v ∈ (R2, µ) | µ(v) = π ,

V hy = w ∈ (R2, µ) | µ(w) > π .




Theorem 9.3.11 Let (R, µ) be a Smarandache plane. If V el ∅ and V hy ∅ , then V eu ∅.

Proof By assumption, we can choose points u ∈ V el and v ∈ V hy. Consider points on

line segment uv in (R2, µ). Notice that µ(u) < π and µ(v) > π. Applying the connectedness

of µ, there exists at least one point w, w ∈ uv such that µ(w) = π, i.e., w ∈ V eu by the

intermediate value theorem on continuous function. Thus V eu ∅.

Corollary 9.3.2 Let (R, µ) be a Smarandache plane. If V eu = ∅ , then either all points of

(R2, µ) are elliptic or hyperbolic.

Corollary 9.3.2 enables one to classify Smarandache planes into classes following:

Euclidean Type. These Smarandache planes in which each point is Euclidean.

Elliptic Type. These Smarandache planes in which each point is elliptic.

Hyperbolic Type. These Smarandache planes in which each point is hyperbolic.

Smarandachely Type. These Smarandache planes in which there are elliptic, Eu-

clidean and hyperbolic points simultaneously. This type can be further classified into

three classes by Corollary 9.3.2:

(S1) Such Smarandache planes just containing elliptic and Euclidean points;

(S2) Such Smarandache planes just containing Euclidean and hyperbolic points;

(S3) Such Smarandache planes containing elliptic, Euclidean and hyperbolic points.

By definition, these isometries of a Euclidean plane R2, i.e., translation, rotation and

reflection exist also in Smarandache planes (R2, µ) of elliptic and hyperbolic types if we

let µ : R2 → [0, π) be a constant< π or > π. We concentrate our discussion on these

Smarandachely types.

X X X X

X X X X

X X X X

a

b

Fig.9.3.6




For convenience, we respectively colour the elliptic, Euclidean and hyperbolic points

by colors red (R), yellow (Y) and white (W). For the cases (S1) or (S2), if there is an

isometry of translation T a,b on (R2, µ), then this Smarandache plane can be only the case

shown in Fig.9.3.6, where X =R or W and all other points colored by Y. Whence, if thereis also a rotation Rθ on (R2, µ), there must be a = b and θ = π/2 with center at X or

the center of one square. In this case, w can easily find a reflection F in a horizontal or a

vertical line passing through X. Whence, there are isometries of types translation, rotation

and reflection in cases (S1) and (S2).

O X U Z

X

U

Z

X

U

Z

XUZ

Fig.9.3.7

Furthermore, if there is an isometry of rotation Rθ on (R2, µ), then this Smarandache

plane can be only the case shown in Fig.9.3.7, where X , U , Z ∈ R, W and all other

points colored by Y. In this case, there are reflections F in lines passing through points O,

X and there are translations T a,b on (R2, µ) only if θ = π/2 and a = b.

X X X X

X X X X

X X X X

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

a

a

Fig.9.3.8



Sec.9.4 Isometries of Pseudo-Euclidean Spaces 333

Consider the case of (S3). In this case, if there is an isometry of translation T a,b

on (R2, µ), then this Smarandache plane can be only the case shown in Fig.9.3.8, where

X ∈ R, W, Z ∈ R, W \ X and all other points colored by Y. Now if there is an

isometry of rotation Rθ on (R2

, µ), there must be a = b and θ = π/2 centered at X, Z orthe center of one square.

Similarly, if there is an isometry of rotation Rθ on (R2, µ) such as those shown in

Fig.9.3.7. Then there are reflections F in lines passing through points O, X. In this case,

there exist translations T a,b on (R2, µ) only if θ = π/2 and a = b.

Summarizing up all the previous discussions, we get the following result on isome-

tries of Smarandache planes (R2, µ) with a continuous function µ : R2 → [0, 2π).

Theorem 9.3.12 Let (R2, µ) be a Smarandachely type plane with µ : R2

→[0, 2π) a

continuous function. Then there are isometries of translation T a,b and rotations Rθ only

if a = b and θ = π/2 , and there are indeed such a Smarandache plane (R2, µ) with

isometries of types translation, rotation and reflection concurrently in each of classes

(S1)-(S3).

§9.4 ISOMETRIES OF PSEUDO-EUCLIDEAN SPACES

9.4.1 Euclidean Space. A Euclidean space on a real vector space E over a field F is a

mapping

· · : E × E → R with (e1, e2) → e1, e2 for ∀e1, e2 ∈ E

such that for e, e1, e2 ∈ E, α ∈ F

( A1) e, e1 + e2 = e, e1 + e, e2;

( A2) e, αe1 = α e, e1;

( A3) e1, e2 = e2, e1;

( A4) e, e ≥ 0 and e, e = 0 if and only if e = 0.In an Euclidean space E, the number

e, e is called its norm, denoted by e for

abbreviation. It can be shown that

(1)

0, e

=

e, 0

= 0 for ∀e ∈ E;

(2)

n

i=1

xie1i ,

m j=1

yie2 j

=

ni=1

mi=1

xi y j

e

1i , e

2 j

, for e

si ∈ E, where 1 ≤ i ≤ maxm, n and






for integers 1 ≤ i, j ≤ n, i j.

Theorem 9.4.2 Let E be an n-dimensional Euclidean space with normal basis ǫ 1, ǫ 2,

· · ·, ǫ n

and T a linear transformation on E determined by Y

t = ai jn

×n

X t , where X =

(ǫ 1, ǫ 2, · · · , ǫ n) and Y = (T (ǫ 1), T (ǫ 2), · · · , T (ǫ n)). Then T is a linear isometry on E if and

only if ai j

n×n

is an orthogonal matrix.

Proof If T is a linear isometry on E, then

T (ǫ i), T (ǫ j)

=

ǫ i, ǫ j

= δi j. Thus

ai1a j1 + ai2a j2 + · · · + aina jn = δi j,

i.e.,ai j

n×n

is an orthogonal matrix by definition.

On the other hand, if

ai j

n×n

is an orthogonal matrix, then we are easily know that

T (ǫ 1), T (ǫ 2),

· · ·, T (ǫ n)

is a normal basis of E. Let b = b1ǫ 1 + b2ǫ 2 +

· · ·+ bnǫ n

∈E. Then

T (b) = T (b1ǫ 1 + b2ǫ 2 + · · · + bnǫ n) = b1T (ǫ 1) + b2T (ǫ 2) + · · · + bnT (ǫ n).

Thus T (b), T (b)

= b2

1 + b22 + · · · + b2

n =

b, b

,

i.e., T is a linear isometry by definition.

9.4.3 Isometry on Euclidean Space. Let E be an n-dimensional Euclidean space with

normal basis ǫ 1, ǫ 2, · · · , ǫ n. As in the case of R2 by the distance-preserving property, any

isometry on E is a composition of three isometries on E following:

Translation Te. A mapping that moves every point ( x1, x2, · · · , xn) of E by

T e : ( x1, x2, · · · , xn) → ( x1 + e1, x2 + e2, · · · , xn + en),

where e = (e1, e2, · · · , en).

Rotation Rθ . A mapping that moves every point of E through a fixed angle about a

fixed point. Similarly, taking the center O to be the origin of polar coordinates (r , φ1, φ2,

· · · , φn−1), a rotation Rθ 1,θ 2,···,θ n−1 : E → E is

Rθ 1,θ 2,···,θ n−1: (r , φ1, φ2, · · · , φn1

) → (r , φ1 + θ 1, φ2 + θ 2, · · · , φn1+ θ n−1),

where θ i is a constant angle, θ i ∈ R (mod2π) for integers 1 ≤ i ≤ n − 1.

Reflection F. A reflection F is a mapping that moves every point of E to its mirror-

image in a fixed Euclidean subspace E ′ of dimensional n−1, denoted by F = F ( E ′). Thus










Now if an s-line L passing through non-Euclidean points x1, x2, · · · , xm ∈ R2, then Theo-

rem 9.4.6 implies that

cos θ x1

sin θ x1

sin θ x1− cos θ x1

cos θ x2

sin θ x2

sin θ x2− cos θ x2

· · · cos θ xm

sin θ xm

sin θ xm− cos θ xm

= I n×n.

Thus

µ( x) =

cos(θ x1+ θ x2

+ · · · + θ xm) sin(θ x1

+ θ x2+ · · · + θ xm

)

sin(θ x1+ θ x2

+ · · · + θ xm) cos(θ x1

+ θ x2+ · · · + θ xm

)

= I n×n.

Whence, θ x1 + θ x2 + · · · + θ xm= 2k π for an integer k . This fact is in agreement with that

of Theorem 9.3.5.

An embedded graph G on Rn is a 1 − 1 mapping τ : G → Rn such that for ∀e, e′ ∈ E (G), τ(e) has no self-intersection and τ(e), τ(e′) maybe only intersect at their end points.

Such an embedded graph G in Rn is denoted by GRn . For example, the n-cube Cn is such

an embedded graph with vertex set V (Cn) = ( x1, x2, · · · , xn) | xi = 0 or 1 f or 1 ≤ i ≤ n and two vertices ( x1, x2, · · · , xn)) and ( x′

1, x′

2, · · · , x′

n) are adjacent if and only if they are

diff er exactly in one entry. We present two n-cubes in Fig.9.4.2 for n = 2 and n = 3.

(0,0) (0,1)

(1,1)(1,0)

n = 2

(0,0,0) (0,0,1)

(0,1,0)

(1,0,0)

(0,1,1)

(1,0,1)

(1,1,1)(1,1,0)

n = 3

Fig.9.4.2

An embedded graph GRn is called Smarandachely if there exists a pseudo-Euclidean

space (Rn, µ) with a mapping µ : x ∈ Rn → x

such that all of its vertices are non-

Euclidean points in (Rn, µ). Certainly, these vertices of valency 1 is not important for

Smarandachely embedded graphs. We concentrate our attention on embedded 2-connected

graphs.



340 Chap.9 Isometries of Smarandache Geometry

Theorem 9.4.7 An embedded 2-connected graph GRn is Smarandachely if and only if

there is a mapping µ : x ∈ Rn → x

and a directed circuit-decomposition

E 12 =

s

i=1

E (−→C i)

such that these matrix equations x∈V (

−→C i)

X x = I n×n 1 ≤ i ≤ s

are solvable.

Proof By definition, if GRn is Smarandachely, then there exists a mapping µ : x ∈

Rn

→ x on Rn

such that all vertices of GRn are non-Euclidean in (Rn

, µ). Notice there areonly two orientations on an edge in E (GRn ). Traveling on GRn beginning from any edge

with one orientation, we get a closed s-line−→C , i.e., a directed circuit. After we traveled

all edges in GRn with the possible orientations, we get a directed circuit-decomposition

E 12

=

si=1

E (−→C i)

with an s-line−→C i for integers 1 ≤ i ≤ s. Applying Theorem 9.4.6, we get

x∈V (

−→C i)

µ( x) = I n×n 1 ≤ i ≤ s.

Thus these equations x∈V (

−→C i)

X x = I n×n 1 ≤ i ≤ s

have solutions X x = µ( x) for x ∈ V (−→C i).

Conversely, if these is a directed circuit-decomposition

E 12

=s

i=1

E (−→C i)

such that these matrix equations x∈V (

−→C i)

X x = I n×n 1 ≤ i ≤ s




are solvable, let X x = A x be such a solution for x ∈ V (−→C i), 1 ≤ i ≤ s. Define a mapping

µ : x ∈ Rn → x

on Rn by

µ( x) = A x if x ∈ V (GRn ), I n×n if x V (GRn ).

Then we get a Smarandachely embedded graph GRn in the pseudo-Euclidean space (Rn, µ)

by Theorem 9.4.6.

Now let C(t ) = ( x1(t ), x2(t ), · · · , xn(t )) be a curve in Rn, i.e.,

C(t ) = x1(t )ǫ 1 + x2(t )ǫ 2 + · · · + xn(t )ǫ n.

If it is an s-line in a pseudo-Euclidean space (Rn

, µ), then

µ(ǫ 1) =x1(t )

| x1(t )|ǫ 1, µ(ǫ 2) =x2(t )

| x2(t )|ǫ 2, · · · , µ(ǫ n) =xn(t )

| xn(t )|ǫ n.

Whence, we get the following result.

Theorem 9.4.8 A curve C(t ) = ( x1(t ), x2(t ), · · · , xn(t )) with parameter t in Rn is an s-line

of a pseudo-Euclidean space (Rn, µ) if and only if

µ(t ) = x1(t )

x2(t ) O

O. . .

xn(t )

.

9.4.5 Isometry on Pseudo-Euclidean Space. We have known Isom(Rn) =Te,Rθ ,F

.

An isometry τ of a pseudo-Euclidean space (Rn, µ) isan isometry on Rn such that µ(τ( x)) =

µ( x) for∀ x ∈ Rn. Clearly, all such isometries form a group Isom(Rn, µ) under composition

operation with Isom(Rn, µ) ≤ Isom(Rn). We determine isometries of pseudo-Euclidean

spaces in this subsection.Certainly, if µ( x) is a constant matrix [c] for ∀ x ∈ Rn, then all isometries on Rn is

also isometries on (Rn, µ). Whence, we only discuss those cases with at least two values

for µ : x ∈ Rn → x

similar to that of (R2, µ).

Translation. Let (Rn, µ) be a pseudo-Euclidean space with an isometry of transla-

tion T e, where e = (e1, e2, · · · , en) and P, Q ∈ (Rn, µ) a non-Euclidean point, a Euclidean




point, respectively. Then µ(T k e

(P)) = µ(P), µ(T k e

(Q)) = µ(Q) for any integer k ≥ 0 by

definition. Consequently,

P, T e(P), T 2

e

(P),· · ·

, T k

e

(P),· · ·

,

Q, T e(Q), T 2e (Q), · · · , T k

e(Q), · · ·

are respectively infinite non-Euclidean and Euclidean points. Thus there are no isometries

of translations if (Rn, µ) is finite.

In this case, if there are rotations Rθ 1,θ 2,···,θ n−1, then there must be θ 1, θ 2, · · · , θ n−1 ∈

0, π/2 and if θ i = π/2 for 1 ≤ i ≤ l, θ i = 0 if i ≥ l + 1, then e1 = e2 = · · · = el+1.

Rotation. Let ( Rn, µ) be a pseudo-Euclidean space with an isometry of rotation

Rθ 1,θ 2,

···,θ n

−1

and P, Q

∈(Rn, µ) a non-Euclidean point, a Euclidean point, respectively. Then

µ( Rθ 1,θ 2,···,θ n−1(P)) = µ(P), µ( Rθ 1 ,θ 2,···,θ n−1

(Q)) = µ(Q) for any integer k ≥ 0 by definition.

Whence,

P, Rθ 1,θ 2,···,θ n−1(P), R2

θ 1,θ 2,···,θ n−1(P), · · · , Rk

θ 1,θ 2,···,θ n−1(P), · · · ,

Q, Rθ 1,θ 2,···,θ n−1(Q), R2

θ 1,θ 2,···,θ n−1(Q), · · · , Rk

θ 1,θ 2,···,θ n−1(Q), · · ·

are respectively non-Euclidean and Euclidean points.

In this case, if there exists an integer k such that θ i|2k π for all integers 1 ≤ i ≤

n − 1, then the previous sequences is finite. Thus there are both finite and infinite pseudo-Euclidean space (Rn, µ) in this case. But if there is an integer i0, 1 ≤ i0 ≤ n − 1 such

that θ i0| 2k π for any integer k , then there must be either infinite non-Euclidean points or

infinite Euclidean points. Thus there are isometries of rotations in a finite non-Euclidean

space only if there exists an integer k such that θ i|2k π for all integers 1 ≤ i ≤ n − 1.

Similarly, an isometry of translation exists in this case only if θ 1, θ 2, · · · , θ n−1 ∈ 0, π/2.

Reflection. By definition, a reflection F in a subspace E ′ of dimensional n − 1 is an

involution, i.e., F 2 = 1Rn . Thus if (Rn, µ) is a pseudo-Euclidean space with an isometry

of reflection F in E ′ and P, Q ∈ (Rn, µ) are respectively a non-Euclidean point and aEuclidean point. Then it is only need that P, F (P) are non-Euclidean points and Q, F (Q)

are Euclidean points. Therefore, a reflection F can be exists both in finite and infinite

pseudo-Euclidean spaces (Rn, µ).

Summing up all these discussions, we get results following for finite or infinite

pseudo-Euclidean spaces.




Theorem 9.4.9 Let (Rn, µ) be a finite pseudo-Euclidean space. Then there maybe isome-

tries of translations T e , rotations Rθ and reflections on (Rn, µ). Furthermore,

(1) If there are both isometries T e and Rθ , where e = (e1, e2,

· · ·, en) and θ =

(θ 1, θ 2, · · · , θ n−1) , then θ 1, θ 2, · · · , θ n−1 ∈ 0, π/2 and if θ i = π/2 for 1 ≤ i ≤ l, θ i = 0

if i ≥ l + 1 , then e1 = e2 = · · · = el+1.

(2) If there is an isometry Rθ 1,θ 2,···,θ n−1 , then there must be an integer k such that θ i | 2k π

for all integers 1 ≤ i ≤ n − 1.

(3) There always exist isometries by putting Euclidean and non-Euclidean points

x ∈ Rn with µ( x) constant on symmetric positions to E ′ in (Rn, µ).

Theorem 9.4.10 Let (Rn, µ) be a infinite pseudo-Euclidean space. Then there maybe

isometries of translations T e , rotations Rθ and reflections on (Rn, µ). Furthermore,

(1) There are both isometries T e and Rθ with e = (e1, e2, · · · , en) and θ = (θ 1, θ 2,

· · · , θ n−1) , only if θ 1, θ 2, · · · , θ n−1 ∈ 0, π/2 and if θ i = π/2 for 1 ≤ i ≤ l, θ i = 0 if i ≥ l + 1 ,

then e1 = e2 = · · · = el+1.

(2) There exist isometries of rotations and reflections by putting Euclidean and non-

Euclidean points in the orbits x Rθ and y

F with a constant µ( x) in (Rn, µ).

We determine isometries on (R3, µ) with a 3-cube C3 shown in Fig.9.4.2. Leta

be

an 3 × 3 orthogonal matrix,

a

I 3×3 and let µ( x1, x2, x3) =

a

for x1, x2, x3 ∈ 0, 1,

otherwise, µ( x1, x2, x3) = I 3×3. Then its isometries consist of two types following:

Rotations:

R1, R2, R3: these rotations through π/2 about 3 axes joining centres of opposite

faces;

R4, R5, R6, R7, R8, R9: these rotations through π about 6 axes joining midpoints of

opposite edges;

R10, R11, R12, R13: these rotations through about 4 axes joining opposite vertices.

Reflection F : the reflection in the centre fixes each of the grand diagonal, reversingthe orientations.

Then Isom(R3, µ) = Ri, F , 1 ≤ i ≤ 13 ≃ S 4 × Z 2. But if letb

be another 3 × 3

orthogonal matrix,ba

and define µ( x1, x2, x3) =a

for x1 = 0, x2, x3 ∈ 0, 1,

µ( x1, x2, x3) =b

for x1 = 1, x2, x3 ∈ 0, 1 and µ( x1, x2, x3) = I 3×3 otherwise. Then only

the rotations R, R2, R3, R4 through π/2, π, 3π/2 and 2π about the axis joining centres of




opposite face

(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1) and (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1),

and reflection F through to the plane passing midpoints of edges

(0, 0, 0) − (0, 0, 1), (0, 1, 0) − (0, 1, 1), (1, 0, 0) − (1, 0, 1), (1, 1, 0) − (1, 1, 1)

or (0, 0, 0) − (0, 1, 0), (0, 0, 1) − (0, 1, 1), (1, 0, 0) − (1, 1, 0), (1, 0, 1) − (1, 1, 1)

are isometries on (R3, µ). Thus Isom(R3, µ) = R1, R2, R3, R4, F ≃ D8.

Furthermore, letai

, 1 ≤ i ≤ 8 be orthogonal matrixes distinct two by two and de-

fine µ(0, 0, 0) =a1

, µ(0, 0, 1) =

a2

, µ(0, 1, 0) =

a3

, µ(0, 1, 1) =

a4

, µ(1, 0, 0) =

a5

,

µ(1, 0, 1) =a6

, µ(1, 1, 0) =

a7

, µ(1, 1, 1) =

a8

and µ( x1, x2, x3) = I 3×3 if x1, x2, x3 0

or 1. Then Isom(R3, µ) is nothing but a trivial group.

§9.5 REMARKS

9.5.1 The Smarandache geometry is proposed by Smarandache by denial the 5th postu-

late for parallel lines in Euclidean postulates on geometry in 1969 (See [Sma1]-[Sma2]

for details). Then a formal definition on such geometry was suggested by Kuciuk and An-

tholy in [KuA1]. More materials and results on Smarandache geometry can be found in

references, such as those of [Sma1]-[Sma2], [Iser1]-[Iser2], [Mao4], [Mao25] and [Liu4].

9.5.2 For Smarandache 2-manifolds, Iseri constructed 2-manifolds by equilateral triangu-

lar disks on Euclidean plane R2. Such manifold can be really come true by paper model in

R3 for elliptic, Euclidean and hyperbolic cases ([Isei1]). Observing the essence of identifi-

cation 5, 6, 7 equilateral triangles in Iseri’s manifolds is in fact a mapping µ : R2 → 5π/3,

2π or 7π/3, a general construction for Smarandache 2-manifolds, i.e., map geometry was

suggested in [Mao3] by applying a general mapping µ : R2 → [0, 2π) on vertices of a

map, and then proved such approach can be used for constructing paradoxist geometry,

anti-geometry and counter-geometry in [Mao4]. It should be noted that a more general

Smarandache n-manifold, i.e., combinatorial manifold was combinatorially constructed

in [Mao15]. Moreover, a diff erential theory on such manifold was also established in

[Mao15]-[Mao17], which can be also found in the surveying monograph [Mao25].

9.5.3 All points are equal in status in a Euclidean space E. But it is not always true in



Sec.9.5 Remarks 345

Smarandache 2-manifolds and pseudo-Euclidean spaces. This fact means that not every

isometry of Rn is still an isometry of (Rn, µ). For finite Smarandache 2-manifolds or

pseudo-Euclidean space, we can determine isometries by a combinatorial approach, i.e.,

maps on surfaces or embedded graphs in Euclidean spaces. But for infinite Smarandache2-manifolds or pseudo-Euclidean spaces, this approach is not always eff ective. However,

we have know all isometries of Euclidean spaces. Applying the fact that every isometry

of a pseudo-Euclidean space (Rn, µ) must be that of Rn, It is not hard for determining

isometries of a pseudo-Euclidean space (Rn, µ).

9.5.4 Let D : E → E be a mapping on a Euclidean space E. If

D( x) − D() y = x − y

holds for all x, y ∈ E, then D is called a norm-preserving mapping. Notice that Theorems

9.4.3 and 9.4.4 is established on the condition of distance-preserving. Whence, They are

also true for norm-preserving mapping, i.e., there exist a orthogonal matrixai j

n×n

, a

constant vector e and a constant number λ such that

G = λai j

n×n

+ e.

9.5.5 Let E be a Euclidean space and T : E → E be a linear mapping. If there exists a

real number λ such that

T (v1), T (v2) = λ2 v1, v2 ,

for all v1, v2 ∈ E, then T is called a linear conformal mapping. It is easily to verify that

T (v) = |λ|v

for v ∈ b f E . Such a linear conformal mapping T is indeed an angle-preserving mapping.

In fact, let v1, v2 be two vectors with angle θ . Then by definition

cos∠(T (v1), T (v2)) =T (v1), T (v2)T (v1) T (v2) =

λ2 v1, v2λ2v1 v2 =

v1, v2v1 v2 = cos θ.

Thus ∠(T (v1), T (v2)) = θ for 0 ≤ ∠(T (v1), T (v2)), θ ≤ π.

Problem 9.5.1 Determine linear conformal mappings on finite or infinite pseudo-Euclidean

spaces (Rn, µ).




9.5.6 For a Euclidean spaces E, a homeomorphism f : E → E is called a di ff erentiable

isometry or conformal di ff erentiable mapping if there is an real number λ such that

d f (v1), d f (v2)

=

v1, v2

or

d f (v1), d f (v2)

= λ2

v1, v2

for ∀ v1, v2 ∈ E. Then it is clear that the integral of a linear isometry is a diff eren-

tiable. and that of a linear conformal mapping is a diff erentiable conformal mapping by

definition. Thus the diff erentiable isometry or conformal diff erentiable mapping is a gen-

eralization of that linear isometry or linear conformal mapping, respectively. Whence, a

natural question arises on pseudo-Euclidean spaces following.

Problem 9.5.2 Determine all di ff erentiable isometries and conformal di ff erentiable map-

pings on a pseudo-Euclidean space (Rn, µ).



CHAPTER 10.

CC Conjecture

The main trend of modern sciences is overlap and hybrid, i.e., combining dif-ferent fields into one underlying a combinatorial structure. This implies the

importance of combinatorics to modern sciences. As a powerful tool for deal-

ing with relations among objectives, combinatorics mushroomed in the past

century, particularly in catering to the need of computer science and children

games. However, an even more important work for mathematician is to apply

it to other mathematics and other sciences besides just to find combinatorial

behavior for objectives. How can it contributes more to the entirely mathemat-

ical science, not just in various games, but in metric mathematics? What is a

right mathematical theory for the original face of our world ? I have brought

a heartening conjecture for advancing mathematics in 2005, i.e., A mathemat-

ical science can be reconstructed from or made by combinatorialization after

a long time speculation on combinatorics, also a bringing about Smarandache

multi-space for mathematics. This conjecture is not just like an open prob-

lem, but more like a deeply thought for advancing the modern mathematics.

i.e., the mathematical combinatorics resulting in the combinatorial conjecture

for mathematics. For example, maps and graphs embedded on surfaces con-

tribute more and more to other branch of mathematics and sciences discussed

in Chapters 1 − 8.



348 Chap.10 CC Conjecture

§10.1 CC CONJECTURE ON MATHEMATICS

10.1.1 Combinatorial Speculation. Modern science has so advanced that to find a

universal genus in the society of sciences is nearly impossible. Thereby a scientist can

only give his or her contribution in one or several fields. The same thing also happens for

researchers in combinatorics. Generally, combinatorics deals with twofold:

Question 1.1. to determine or find structures or properties of configurations, such as those

structure results appeared in graph theory, combinatorial maps and design theory,..., etc..

Question 1.2. to enumerate configurations, such as those appeared in the enumeration of

graphs, labeled graphs, rooted maps, unrooted maps and combinatorial designs,...,etc..

Consider the contribution of a question to science. We can separate mathematical

questions into three ranks:

Rank 1 they contribute to all sciences.

Rank 2 they contribute to all or several branches of mathematics.

Rank 3 they contribute only to one branch of mathematics, for instance, just to the graph

theory or combinatorial theory.

Classical combinatorics is just a rank 3 mathematics by this view. This conclusion

is despair for researchers in combinatorics, also for me 5 years ago. Whether can combi-natorics be applied to other mathematics or other sciences? Whether can it contributes

to human’s lives, not just in games?

Although become a universal genus in science is nearly impossible, our world is a

combinatorial world . A combinatorician should stand on all mathematics and all sciences,

not just on classical combinatorics and with a real combinatorial notion, i.e., combine

di ff erent fields into a unifying field , such as combine diff erent or even anti-branches in

mathematics or science into a unifying science for its freedom of research. This notion

requires us answering three questions for solving a combinatorial problem before. What

is this problem working for? What is its objective? What is its contribution to science or

human’s society? After these works be well done, modern combinatorics can applied to

all sciences and all sciences are combinatorialization.

10.1.2 CC Conjecture. There is a prerequisite for the application of combinatorics

to other mathematics and other sciences, i.e, to introduce various metrics into combina-



Sec.10.1 CC Conjecture on Mathematics 349

torics, ignored by the classical combinatorics since they are the fundamental of scientific

realization for our world. For applying combinatorics to other branch of mathematics, a

good idea is to pullback measures on combinatorial objects again, ignored by the classical

combinatorics and reconstructed or make combinatorial generalization for the classicalmathematics, such as those of algebra, Euclidean geometry, diff erential geometry, Rie-

mann geometry, metric geometries, · · · and the mechanics, theoretical physics, · · ·. This

notion naturally induces the combinatorial conjecture for mathematics, abbreviated to CC

conjecture following.

Conjecture 10.1.1(CC Conjecture) The mathematical science can be reconstructed from

or made by combinatorialization.

Remark 10.1.1 We need some further clarifications for this conjecture.

(1) This conjecture assumes that one can select finite combinatorial rulers and ax-

ioms to reconstruct or make generalization for classical mathematics.

(2) The classical mathematics is a particular case in the combinatorialization of

mathematics, i.e., the later is a combinatorial generalization of the former.

(3) We can make one combinatorialization of diff erent branches in mathematics and

find new theorems after then.

Therefore, a branch in mathematics can not be ended if it has not been combinato-

rialization and all mathematics can not be ended if its combinatorialization has not com-

pleted. There is an assumption in one’s realization of our world, i.e., science can be made

by mathematicalization, which enables us get a similar combinatorial conjecture for the

science.

Conjecture 10.1.2(CCS Conjecture) Science can be reconstructed from or made by com-

binatorialization.

A typical example for the combinatorialization of classical mathematics is the com-

binatorial surface theory, i.e., a combinatorial theory for surfaces discussed in Chapter 4.

Combinatorially, a surface S is topological equivalent to a polygon with even number of

edges by identifying each pairs of edges along a given direction on it. If label each pair of

edges by a letter e, e ∈ E, a surface S is also identifying to a cyclic permutation such that

each edge e, e ∈ E just appears two times in S , one is e and another is e−1. Let a, b, c, · · ·denote the letters in E and A, B, C , · · · the sections of successive letters in a linear order on




a surface S (or a string of letters on S ). Then, a surface can be represented as follows:

S = (· · · , A, a, B, a−1, C , · · ·),

where, a ∈ E, A, B, C denote a string of letters. Define three elementary transformations

as follows:

(O1) ( A, a, a−1, B) ⇔ ( A, B);

(O2) (i) ( A, a, b, B, b−1, a−1) ⇔ ( A, c, B, c−1);

(ii) ( A, a, b, B, a, b) ⇔ ( A, c, B, c);

(O3) (i) ( A, a, B, C , a−1, D) ⇔ ( B, a, A, D, a−1, C );

(ii) ( A, a, B, C , a, D) ⇔ ( B, a, A, C −1, a, D−1).

If a surface S can be obtained from S 0 by these elementary transformations O1-O3,we say that S is elementary equivalent with S 0, denoted by S ∼ El S 0. Then we can get

the classification theorem of compact surface as follows:

Any compact surface S is homeomorphic to one of the following standard surfaces:

(P0) the sphere: aa−1;

(Pn) the connected sum of n, n ≥ 1 tori:

a1b1a−11 b−1

1 a2b2a−12 b−1

2 · · · anbna−1n b−1

n ;

(Qn) the connected sum of n, n ≥ 1 projective planes:

a1a1a2a2 · · · anan.

We have known what is a map in Chapter 5. By the view of combinatorial maps,

these standard surfaces P0, Pn, Qn for n ≥ 1 is nothing but the bouquet Bn on a locally

orientable surface with just one face. Therefore, the maps are nothing but the combinato-

rialization of surfaces.

10.1.3 CC Problems in Mathematics. Many open problems are motivated by the CC

conjecture. Here we present some of them.

Problem 10.1.1 Simple-Connected Riemann Surface. The uniformization theorem on

simple connected Riemann surfaces is one of those beautiful results in Riemann surfaces

stated as follows ([FaK1]).



Sec.10.1 CC Conjecture on Mathematics 351

Theorem 10.1.1 If S is a simple connected Riemann surface, then S is conformally

equivalent to one and only one of the following three:

(1)

C∞;

(2) C;

(3) = z ∈ C|| z| < 1.

We have proved in Chapter 5 that any automorphism of map is conformal. Therefore, we

can also introduced the conformal mapping between maps. Then, how can one define the

conformal equivalence for maps enabling us to get the uniformization theorem of maps?

What is the correspondence class maps with the three type (1)-(3) Riemann surfaces?

Problem 10.1.2 Riemann-Roch Theorem. Let S be a Riemann surface. A divisor on

S is a formal symbol

U =

k i=1

Pα(Pi)i

with Pi ∈ S, α(Pi) ∈ Z. Denote by Div(S) the free commutative group on the points in Sand define

degU =

k i=1

α(Pi).

Denote by H (S) the field of meromorphic function on S. Then for ∀ f ∈ H (S) \ 0, f

determines a divisor ( f )

∈Div(

S) by

( f ) =P∈S

Pord P f ,

where, if we write f ( z) = zng( z) with g holomorphic and non-zero at z = P, then the

ord P f = n. For U 1 =

P∈SPα1(P), U 2 =

P∈S

Pα2(P), ∈ Div(S), call U 1 ≥ U 2 if α1(P) ≥α2(P). Now we define a vector space

L(U ) = f ∈ H (S)|( f ) ≥ U , U ∈ Div(S)

Ω(U ) = ω|ω is an abelian di f f erential with (ω) ≥ U.

Then the Riemann-Roch theorem says that([WLC1])

dim( L(U −1)) = degU − g(S) + 1 + dimΩ(S).

Comparing with the divisors and their vector space, there ia also cycle space and cocycle

space in graphical space theory ([Liu1]). Then what is their relation? whether can one

rebuilt the Riemann-Roch theorem by maps, i.e., find its discrete form?




Problem 10.1.3 Combinatorial Construction of Algebraic Curve. A complex plane

algebraic curve Cl is a homogeneous equation f ( x, y, z) = 0 in P2C = (C 2 \ (0, 0, 0))/ ∼,

where f ( x, y, z) is a polynomial in x, y and z with coefficients in C. The degree of f ( x, y, z)

is defined to be the degree of the curve Cl. For a Riemann surface S , a well-known result isthat ([WSY1]) there is a holomorphic mapping ϕ : S → P2C such that ϕ(S ) is a complex

plane algebraic curve and

g(S ) =(d (ϕ(S )) − 1)(d (ϕ(S )) − 2)

2.

By definition, we have known that a combinatorial map is on surface with genus. Then

whether can one get an algebraic curve by all edges in a map or by make operations on

the vertices or edges of the map to get plane algebraic curve with given k-multiple points?

and then how do one find the equation f ( x, y, z) = 0?

Problem 10.1.4 Classification of s-Manifolds by Map. We have classified the closed

s-manifolds by maps in the last chapter. For the general s-manifolds, their correspon-

dence combinatorial model is the map on surfaces with boundary, founded by Bryant and

Singerman in 1985. The later is also related to that of modular groups of spaces and need

to investigate further itself. Now the questions are

(1) How can one combinatorially classify the general s-manifolds by maps with

boundary?

(2) How can one find the automorphism group of an s-manifold?

(3) How can one know the numbers of non-isomorphic s-manifolds, with or without

roots?

(4) Find rulers for drawing an s-manifold on surface, such as, the torus, the projec-

tive plane or Klein bottle, not just the plane.

These s-manifolds only apply such triangulations of surfaces with vertex valency in

5, 6, 7. Then what is its geometrical meaning of other maps, such as, 4-regular maps on

surfaces. It is already known that the later is related to the Gauss cross problem of curves

([Liu1]).

Problem 10.1.5 Gauss Mapping. In the classical diff erential geometry, a Gauss map-

ping among surfaces is defined as follows([Car1]):

Definition 10.1.1 Let S ⊂ R3 be a surface with an orientation N. The mapping N : S →



Sec.10.2 CC Conjecture To Mathematics 353

R3 takes its value in the unit sphere

S 2 = ( x, y, z) ∈ R3| x2 + y2 + z2 = 1

along the orientation N. The map N : S → S 2 , thus defined, is called the Gauss mapping.

We know that for a point P ∈ S such that the Gaussian curvature K (P) 0 and V a

connected neighborhood of P with K does not change sign,

K (P) = lim A→0

N ( A)

A,

where A is the area of a region B ⊂ V and N ( A) is the area of the image of B by the Gauss

mapping N : S → S 2. Now the questions are

(1) What is its combinatorial meaning of the Gauss mapping? How to realizes it by

maps?

(2) how we can define various curvatures for maps and rebuilt the results in the

classical di ff erential geometry?

Problem 10.1.6 Gauss-Bonnet Theorem. Let S be a compact orientable surface. Then S

Kd σ = 2πχ(S),

where K is Gaussian curvature on

S. This is the famous Gauss-Bonnet theorem for com-

pact surface ([WLC1], [WSY1]). This theorem should has a combinatorial form. Now

the questions are

(1) How can one define various metrics for combinatorial maps, such as those of

length, distance, angle, area, curvature, · · ·?(2) Can one rebuilt the Gauss-Bonnet theorem by maps for dimensional 2 or higher

dimensional compact manifolds without boundary?

§10.2 CC CONJECTURE TO MATHEMATICS

10.2.1 Contribution to Algebra. By the view of combinatorics, algebra can be seen

as a combinatorial mathematics itself. The combinatorial speculation can generalize it by

the means of combinatorialization. For this objective, a Smarandachely multi-algebraic

system is combinatorially defined in the following definition.




Definition 10.2.1 For any integers n, n ≥ 1 and i, 1 ≤ i ≤ n, let Ai be a set with an

operation set O( Ai) such that ( Ai, O( Ai)) is a complete algebraic system. Then the union

n

i=1

( Ai, O( Ai))

is called an n multi-algebra system.

An example of multi-algebra systems is constructed by a finite additive group. Now

let n be an integer, Z 1 = (0, 1, 2, · · · , n − 1, +) an additive group (mod n) and P =

(0, 1, 2, · · · , n − 1) a permutation. For any integer i, 0 ≤ i ≤ n − 1, define

Z i+1 = Pi( Z 1)

satisfying that if k + l = m in Z 1, then Pi(k ) +i Pi(l) = Pi(m) in Z i+1, where +i denotes the

binary operation +i : (Pi(k ), Pi(l)) → Pi(m). Then we know thatn

i=1

Z i

is an n multi-algebra system .

The conception of multi-algebra systems can be extensively used for generalizing

conceptions and results for these existent algebraic structures, such as those of groups,

rings, bodies, fields and vector spaces, · · ·, etc.. Some of them are explained in the fol-

lowing.

Definition 10, 2.2 Let G =n

i=1Gi be a closed multi-algebra system with a binary operation

set O(G) = ×i, 1 ≤ i ≤ n. If for any integer i, 1 ≤ i ≤ n, (Gi; ×i) is a group and for

∀ x, y, z ∈ G and any two binary operations¡

×¢

and ¡

¢

, × , there is one operation,

for example the operation × satisfying the distribution law to the operation¡

¢

provided

their operation results existing, i.e.,

x × ( y z) = ( x × y) ( x × z),

( y

z)×

x = ( y×

x)

( z×

x),

then G is called a multi-group.

For a multi-group (G, O(G)), G1 ⊂ G and O(G1) ⊂ O(G), call (G1, O(G1)) a sub-

multi-group of (G, O(G)) if G1 is also a multi-group under the operations in O(G1), de-

noted by G1 G. For two sets A and B, if A

B = ∅, we denote the union A

B by

A

B. Then we get a generalization of the Lagrange theorem on finite group following.




Theorem 10.2.1 For any sub-multi-group H of a finite multi-group G, there is a repre-

sentation set T, T ⊂ G, such that

G = x∈T

x H .

For a sub-multi-group H of G, × ∈ O( H ) and ∀g ∈ G(×), if for ∀h ∈ H ,

g × h × g−1 ∈ H ,

then call H a normal sub-multi-group of G. An order of operations in O(G) is said an

oriented operation sequence, denoted by−→O(G). We get a generalization of the Jordan-

Holder theorem for finite multi-groups following.

Theorem 10.2.2 For a finite multi-group G =n

i=1Gi and an oriented operation sequence

−→O(G) , the length of maximal series of normal sub-multi-groups is a constant, only depen-

dent on G itself.

A complete proof of Theorems 10.2.1 and 10.2.2 can be found in the reference

[Mao6]. Notice that if we choose n = 2 in Definition 10.2.2, G1 = G2 = G. Then Gis a body. If (G1; ×1) and (G2; ×2) both are commutative groups, then G is a field. For

multi-algebra systems with two or more operations on one set, we introduce the concep-

tion of multi-rings and multi-vector spaces in the following.

Definition 10.2.3 Let R =m

i=1 Ri be a closed multi-algebra system with double binary

operation set O( R) = (+i, ×i), 1 ≤ i ≤ m. If for any integers i, j, i j, 1 ≤ i, j ≤ m,

( Ri; +i, ×i) is a ring and for ∀ x, y, z ∈ R,

( x +i y) + j z = x +i ( y + j z), ( x ×i y) × j z = x ×i ( y × j z)

and

x

×i ( y + j z) = x

×i y + j x

×i z, ( y + j z)

×i x = y

×i x + j z

×i x

provided all their operation results exist, then R is called a multi-ring. If for any integer

1 ≤ i ≤ m, ( R; +i, ×i) is a filed, then R is called a multi-filed.

Definition 10.2.4 Let V =k

i=1

V i be a closed multi-algebra system with binary operation

set O(V ) = (+i, ·i) | 1 ≤ i ≤ m and F =k

i=1

F i a multi-filed with double binary operation




set O(F ) = (+i, ×i) | 1 ≤ i ≤ k . If for any integers i, j, 1 ≤ i, j ≤ k and ∀a, b, c ∈ V,

k 1, k 2 ∈ F,

(1) (V i; +i,

·i) is a vector space on F i with vector additive +i and scalar multiplication

·i;

(2) (a+ib)+ jc = a+i(b+ jc);

(3) (k 1 +i k 2) · j a = k 1 +i (k 2 · j a);

provided all those operation results exist, then V is called a multi-vector space on the

multi-filed F with a binary operation set O(V ) , denoted by (V ; F ).

Similarly, we also obtained results for multi-rings and multi-vector spaces to gener-

alize classical results in rings or linear spaces.

10.2.2 Contribution to Metric Space. First, we generalize classical metric spaces by

the combinatorial speculation.

Definition 10.2.5 A multi-metric space is a union M =m

i=1

M i such that each M i is a space

with metric ρi for ∀i, 1 ≤ i ≤ m.

We generalized two well-known results in metric spaces.

Theorem 10.2.3 Let

M =

m

i=1

M i be a completed multi-metric space. For an ǫ -disk se-

quence B(ǫ n, xn) , where ǫ n > 0 for n=

1, 2, 3, · · · , the following conditions hold:

(1) B(ǫ 1, x1) ⊃ B(ǫ 2, x2) ⊃ B(ǫ 3, x3) ⊃ · · · ⊃ B(ǫ n, xn) ⊃ · · ·;(2) lim

n→+∞ǫ n = 0.

Then+∞n=1

B(ǫ n, xn) only has one point.

Theorem 10.2.4 Let M =m

i=1 M i be a completed multi-metric space and T a contraction

on

M. Then

1 ≤#

Φ(T ) ≤ m.

A complete proof of Theorems 10.2.3 and 10.2.4 can be found in the reference

[Mao7]. Particularly, let m = 1. We get the Banach fixed-point theorem again.

Corollary 10.2.1(Banach) Let M be a metric space and T a contraction on M. Then T

has just one fixed point.




A Smarandache n-manifold is an n-dimensional manifold that supports a Smaran-

dache geometry. Now there are many approaches to construct Smarandache manifolds

for n = 2. A general way is by the so called map geometries without or with boundary

underlying orientable or non-orientable maps.

Definition 10.2.6 For a combinatorial map M with each vertex valency≥ 3 , endow with

a real number µ(u), 0 < µ(u) < 4π ρ M (u)

, to each vertex u, u ∈ V ( M ). Call ( M , µ) a

map geometry without boundary, µ(u) an angle factor of the vertex u and orientablle or

non-orientable if M is orientable or not.

Definition 10.2.7 For a map geometry ( M , µ) without boundary and faces f 1, f 2, · · · , f l ∈F ( M ), 1 ≤ l ≤ φ( M )−1 , if S ( M )\ f 1, f 2, · · · , f l is connected, then call ( M , µ)−l = (S ( M )\

f 1, f 2, · · · , f l, µ) a map geometry with boundary f 1, f 2, · · · , f l , where S ( M ) denotes thelocally orientable surface underlying map M.

The realization for vertices u, v, w ∈ V ( M ) in a space R3 is shown in Fig.3.2, where

ρ M (u) µ(u) < 2π for the vertex u, ρ M (v) µ(v) = 2π for the vertex v and ρ M (w) µ(w) > 2π for

the vertex w, are called to be elliptic, Euclidean or hyperbolic, respectively.

u

u

u

ρ M (u) µ(u) < 2π ρ M (u) µ(u) = 2π ρ M (u) µ(u) > 2π

Fig.10.2.1

Theorem 10.2.5 There are Smarandache geometries, including paradoxist geometries,

non-geometries and anti-geometries in map geometries without or with boundary.

A proof of this result can be found in [Mao4]. Furthermore, we generalize the ideas

in Definitions 10.2.6 and 10.2.7 to metric spaces and find new geometries.

Definition 10.2.8 Let U and W be two metric spaces with metric ρ , W ⊆ U. For ∀u ∈ U, if

there is a continuous mapping ω : u → ω(u) , where ω(u) ∈ Rn for an integer n, n ≥ 1 such

that for any number ǫ > 0 , there exists a number δ > 0 and a point v ∈ W, ρ(u − v) < δ




such that ρ(ω(u) − ω(v)) < ǫ , then U is called a metric pseudo-space if U = W or a

bounded metric pseudo-space if there is a number N > 0 such that ∀w ∈ W, ρ(w) ≤ N,

denoted by (U , ω) or (U −, ω) , respectively.

For the case n = 1, we can also explain ω(u) being an angle function with 0 < ω(u) ≤4π as in the case of map geometries without or with boundary, i.e.,

ω(u) =

ω(u)(mod 4π), if u ∈ W,

2π, if u ∈ U \ W (∗)

and get some interesting metric pseudo-space geometries. For example, let U = W =

Euclid plane =

, then we obtained some interesting results for pseudo-plane geometries

(

, ω) as shown in results following ([Mao4]).

Theorem 10.2.6 In a pseudo-plane (, ω) , if there are no Euclidean points, then all

points of (

, ω) is either elliptic or hyperbolic.

Theorem 10.2.7 There are no saddle points and stable knots in a pseudo-plane plane

(

, ω).

Theorem 10.2.8 For two constants ρ0, θ 0 , ρ0 > 0 and θ 0 0 , there is a pseudo-plane

(

, ω) with

ω( ρ, θ ) = 2(π − ρ0

θ 0 ρ) or ω( ρ,θ ) = 2(π +

ρ0

θ 0 ρ)

such that

ρ = ρ0

is a limiting ring in (

, ω).

Now for an m-manifold M m and ∀u ∈ M m, choose U = W = M m in Definition 10.2.8

for n = 1 and ω(u) a smooth function. We get a pseudo-manifold geometry ( M m, ω) on

M m. By definitions , a Minkowski norm on M m is a function F : M m → [0, +∞) such that

(1) F is smooth on M m \ 0;

(2) F is 1-homogeneous, i.e., F (λu) = λF (u) for u ∈ M m

and λ > 0;(3) for ∀ y ∈ M m \ 0, the symmetric bilinear form g y : M m × M m → R with

g y(u, v) =1

2

∂2F 2( y + su + tv)

∂s∂t |t =s=0

is positive definite and a Finsler manifold is a manifold M m endowed with a function

F : T M m → [0, +∞) such that



Sec.10.3 CC Conjecture To Physics 359

(1) F is smooth on T M m \ 0 =T x M m \ 0 : x ∈ M m;

(2) F |T x M m → [0, +∞) is a Minkowski norm for ∀ x ∈ M m.

As a special case, we choose ω( x) = F ( x) for x ∈ M m, then ( M m, ω) is a Finsler

manifold . Particularly, if ω( x) = g x( y, y) = F 2( x, y), then ( M m, ω) is a Riemann mani-

fold . Therefore, we get a relation for Smarandache geometries with Finsler or Riemann

geometry.

Theorem 10.2.9 There is an inclusion for Smarandache, pseudo-manifold, Finsler and

Riemann geometries as shown in the following:

S marandache geometries ⊃ pseudo − mani f old geometries

⊃ Finsler geometry

⊃ Riemann geometry.

§10.3 CC CONJECTURE TO PHYSICS

The progress of theoretical physics in last twenty years of the 20th century enables human

beings to probe the mystic cosmos: where are we came from? where are we going to?.

Today, these problems still confuse eyes of human beings. Accompanying with researchin cosmos, new puzzling problems also arose: Whether are there finite or infinite cos-

moses? Are there just one? What is the dimension of the Universe? We do not even know

what the right degree of freedom in the Universe is, as Witten said.

We are used to the idea that our living space has three dimensions: length, breadth

and height , with time providing the fourth dimension of spacetime by Einstein. Applying

his principle of general relativity, i.e. all the laws of physics take the same form in any

reference system and equivalence principle, i.e., there are no di ff erence for physical e ff ects

of the inertial force and the gravitation in a field small enough., Einstein got the equation

of gravitational field

R µν − 1

2 Rg µν + λg µν = −8πGT µν.

where R µν = Rνµ = Rα µiν,

Rα µiν =

∂Γi µi

∂ xν− ∂Γi

µν

∂ xi+ Γα

µiΓiαν − Γα

µνΓiαi,




Γgmn =

1

2g pq(

∂gmp

∂un+

∂gnp

∂um− ∂gmn

∂u p)

and R = gνµ Rνµ. Combining the Einstein’s equation of gravitational field with the cosmo-

logical principle, i.e., there are no di ff erence at di ff erent points and di ff erent orientations

at a point of a cosmos on the metric 104l. y. , Friedmann got a standard model of cosmos.

The metrics of the standard cosmos are

d s2 = −c2dt 2 + a2(t )[dr 2

1 − Kr 2+ r 2(d θ 2 + sin2 θ d ϕ2)]

and

gtt = 1, grr = − R2(t )

1 − Kr 2, gφφ = −r 2 R2(t )sin2 θ.

The standard model of cosmos enables the birth of big bang model of the Universe

in thirties of the 20th century. The following diagram describes the developing process of

our cosmos in diff erent periods after the big bang.

Fig.4.1

10.3.1 M-Theory. The M-theory was established by Witten in 1995 for the unity of

those five already known string theories and superstring theories, which postulates that

all matter and energy can be reduced to branes of energy vibrating in an 11 dimensional

space, then in a higher dimensional space solve the Einstein’s equation of gravitational




field under some physical conditions. Here, a brane is an object or subspace which can

have various spatial dimensions. For any integer p ≥ 0, a p-brane has length in p di-

mensions. For example, a 0-brane is just a point or particle; a 1-brane is a string and a

2-brane is a surface or membrane, · · ·.We mainly discuss line elements in diff erential forms in Riemann geometry. By a

geometrical view, these p-branes in M-theory can be seen as volume elements in spaces.

Whence, we can construct a graph model for p-branes in a space and combinatorially

research graphs in spaces.

Definition 10.3.1 For each m-brane B of a space Rm , let (n1(B), n2(B), · · · , n p(B)) be its

unit vibrating normal vector along these p directions and q : Rm → R4 a continuous

mapping. Now construct a graph phase (

G, ω, Λ) by

V (G) = p − branes q(B),

E (G) = (q(B1), q(B2))|there is an action between B1 and B2,

ω(q(B)) = (n1(B), n2(B), · · · , n p(B)),

and

Λ(q(B1), q(B2)) = f orces between B1 and B2.

Then we get a graph phase (G, ω, Λ) in R4. Similarly, if m = 11 , it is a graph phase for

the M-theory.

As an example for applying M-theory to find an accelerating expansion cosmos of

4-dimensional cosmoses from supergravity compactification on hyperbolic spaces is the

Townsend-Wohlfarth type metric in which the line element is

ds2 = e−mφ(t )(−S 6dt 2 + S 2dx23) + r 2C e

2φ(t )ds2 H m

,

where

φ(t ) =1

m − 1(ln K (t ) − 3λ0t ),

S 2 = K m

m−1 e− m+2m−1 λ0 t

and

K (t ) =λ0ζ r c

(m − 1) sin[λ0ζ |t + t 1|]




with ζ =√

3 + 6/m. This solution is obtainable from space-like brane solution and if

the proper time ς is defined by d ς = S 3(t )dt , then the conditions for expansion and

acceleration are dSd ς

> 0 and d 2Sd ς 2

> 0. For example, the expansion factor is 3.04 if m = 7,

i.e., a really expanding cosmos.According to M-theory, the evolution picture of our cosmos started as a perfect 11

dimensional space. However, this 11 dimensional space was unstable. The original 11

dimensional space finally cracked into two pieces, a 4 and a 7 dimensional subspaces. The

cosmos made the 7 of the 11 dimensions curled into a tiny ball, allowing the remaining 4

dimensions to inflate at enormous rates, the Universe at the final.

10.3.2 Combinatorial Cosmos. The combinatorial notion made the following combi-

natorial cosmos in the reference.

Definition 10.3.2 A combinatorial cosmos is constructed by a triple (Ω, ∆, T ) , where

Ω =i≥0

Ωi, ∆ =i≥0

Oi

and T = t i; i ≥ 0 are respectively called the cosmos, the operation or the time set with

the following conditions hold.

(1) (Ω, ∆) is a Smarandache multi-space dependent on T , i.e., the cosmos (Ωi, Oi) is

dependent on time parameter t i for any integer i, i≥

0.

(2) For any integer i, i ≥ 0 , there is a sub-cosmos sequence

(S ) : Ωi ⊃ · · · ⊃ Ωi1 ⊃ Ωi0

in the cosmos (Ωi, Oi) and for two sub-cosmoses (Ωi j, Oi) and (Ωil, Oi) , if Ωi j ⊃ Ωil , then

there is a homomorphism ρΩi j,Ωil: (Ωi j, Oi) → (Ωil, Oi) such that

(i) for ∀(Ωi1, Oi), (Ωi2, Oi), (Ωi3, Oi) ∈ (S ) , if Ωi1 ⊃ Ωi2 ⊃ Ωi3 , then

ρΩi1,Ωi3 = ρΩi1,Ωi2 ρΩi2,Ωi3 ,

where¡

¢

denotes the composition operation on homomorphisms.

(ii) for ∀g, h ∈ Ωi , if for any integer i, ρΩ,Ωi(g) = ρΩ,Ωi

(h) , then g = h.

(iii) for ∀i, if there is an f i ∈ Ωi with

ρΩi,Ωi

Ω j

( f i) = ρΩ j,Ωi

Ω j

( f j)




for integers i, j, Ωi

Ω j ∅ , then there exists an f ∈ Ω such that ρΩ,Ωi

( f ) = f i for any

integer i.

By this definition, there is just one cosmos Ω and the sub-cosmos sequence is

R4 ⊃ R3 ⊃ R2 ⊃ R1 ⊃ R0 = P ⊃ R−7 ⊃ · · · ⊃ R−

1 ⊃ R−0 = Q.

in the string / M-theory. In Fig.10.3.2, we have shown the idea of the combinatorial cos-

mos.

Fig.10.3.2

For spaces of dimensional 5 or 6, it has been established a dynamical theory by

combinatorial notion (see [Pap1]-[Pap2] for details). In this dynamics, we look for asolution in the Einstein’s equation of gravitational field in 6-dimensional spacetime with

a metric of the form

ds2 = −n2(t , y, z)dt 2 + a2(t , y, z)d

2k

+b2(t , y, z)dy2 + d 2(t , y, z)dz2

where d 2

k represents the 3-dimensional spatial sections metric with k = −1, 0, 1 respec-

tive corresponding to the hyperbolic, flat and elliptic spaces. For 5-dimensional space-

time, deletes the indefinite z in this metric form. Now consider a 4-brane moving in a

6-dimensional Schwarzschild-ADS spacetime, the metric can be written as

ds2 = −h( z)dt 2 +z2

l2d

2k

+h−1( z)dz2,

where

d

2k

=dr 2

1 − kr 2+ r 2d Ω2

(2) + (1 − kr 2)dy2




and

h( z) = k +z2

l2− M

z3.

Then the equation of a 4-dimensional cosmos moving in a 6-spacetime is

2¨ R

R+ 3(

˙ R

R)2 = −3

κ 4(6)

64ρ2 − κ 4(6)

8ρ p − 3

κ

R2− 5

l2

by applying the Darmois-Israel conditions for a moving brane. Similarly, for the case of

a( z) b( z), the equations of motion of the brane are

d 2d ˙ R − d ¨ R√ 1 + d 2 ˙ R2

−√

1 + d 2 ˙ R2

n(d n ˙ R +

∂ zn

d − (d ∂ zn − n∂ zd ) ˙ R2) = −

κ 4(6)

8(3( p + ρ) + ˆ p),

∂ za

ad

1 + d 2 ˙ R2 = −

κ 4(6)

8( ρ + p − ˆ p),

∂ zbbd 1 + d 2 ˙ R2 = −

κ 4(6)

8( ρ − 3( p − ˆ p)),

where the energy-momentum tensor on the brane is

T µν = hναT α µ − 1

4T h µν

with T α µ = diag(− ρ, p, p, p, ˆ p) and the Darmois-Israel conditions

[K µν] = −κ 2(6)T µν,

where K µν is the extrinsic curvature tensor.

The combinatorial cosmos also presents new questions to combinatorics, such as:

(1) Embed a graph into spaces with dimensional≥ 4;

(2) Research the phase space of a graph embedded in a space;

(3) Establish graph dynamics in a space with dimensional≥ 4, · · ·, etc..

For example, we have gotten the following result for graphs in spaces.

Theorem 10.3.1 A graph G has a nontrivial including multi-embedding on spheres P 1 ⊃P2 ⊃ · · · ⊃ Ps if and only if there is a block decomposition G =

si=1

Gi of G such that for

any integer i, 1 < i < s,

(1) Gi is planar;

(2) for ∀v ∈ V (Gi) , N G( x) ⊆ (i+1

j=i−1V (G j)).

A complete proof of Theorem 10.3.1 can be found in [Mao4]. Further consideration

of combinatorial cosmos will enlarge the knowledge of combinatorics and cosmology,

also get the combinatorialization for cosmological science.



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Index

A

Abelian multigroup 36

Abelian system 7

additive group 96

adjacency matrix 81

affine group 48

affine transformation 48

A J -compatible 221

A J -uniform 221

algebraic system 6

analytic mapping 146

angle on map 227

angle transformation 227

anti-geometry 310

arc 118

arc transitive graph 101

arcwise-connected 118

asymmetric graph 285

associative 7

automorphism 49, 92, 149, 184, 310

automorphism group 49, 92, 149, 186

B

Banach fixed-point theorem 355band decomposition 170

basis 26

basic permutation 175

bigroup 37

bijection 4

binary operation 6

brane 359

Burnside Lemma 42

C

cardinality 4

Cayley graph 92

Cayley map 195

CC conjecture 348

cell embedding 168

center 50, 51

characteristic subgroup 52

circuit 81

combinatorial cosmos 361

commutator subgroup 53

compact 120

complete map 197, 248

conformal diff erentiable mapping 345

conjugate group 51connected sum 125

connectivity 85

continuous mapping 119

coset 13

coset representation 14

countable set 4

counter-projective geometry 309

covering space 212covering mapping 213

crosscap map group 292

curvature 321, 328, 336

cycle index 299

cyclic group 12






hyperbolic vertex 310

hyperbolic tessellation 200

hypergraph 115

I

image 3, 20, 35

imprimitive block 61

imprimitive group 61

inclusion mapping 141

induced action 105

induced subgraph 83

infinite set 4

injection 4

inner automorphism 50

intersection set 2

inverse 7

involution 16

Iseri’s manifold 310

isometry 318, 323

isomorphic graph 82

isomorphic group 10

isomorphic multigroup 31

isomorphic rooted map 272

isomorphism 20, 149, 181

K

kernel 20, 35Klein 4-group 8, 175

Klein surface 146

k -factorable 86

k -transitive 46

k -transitive extended 71

L

Legendre symbol 58

lifted automorphism 220

lifting arc 212

lifting graph 215

lifting map 217

limit point 118

linear conformal mapping 344

linear isometry 333

local action group 69

locally compact 120

locally transitive group 69

locally regular group 69

locally k -transitive group 70

locally sharply k -transitive group 70

locally primitive group 71

loop 80

M

map 176

map geometry 356

map-automorphism graph 260

map exponent 221

map group 186

mapping 3

metric pseudo-space 357

metric space 317minimal normal subgroup 66

morphism 148

multi-algebra system 353

multigroup 30

multiple edge 80



Index 381

multigroup action graph 110

multi-metric space 355

multipolygon 206

multisurface 206

N

natural homomorphism 23

NEC group 150

neighborhood 81, 118

n-manifold 121

non-Euclid area of map 227

non-Euclidean point 321, 335

non-geometry 309

non-orientable genus polynomial 276

non-orientable map 179

non-orientable surface 123

norm 332

norm-preserving mapping 344

normally Cayley graph 93

normalizer 51

normal subgroups 15

normal submultigroup 33

n-sheeted covering 213

O

odd permutation 17

one face map 261one-point extended 71

opened cover 120

order 11, 80

orientable embedding index 269

orientable genus polynomial 276

orientable map 179

orientable surface 123

orientation-preserving 123, 170, 242

orientation-reversing 123, 170, 242

outer automorphism group 51

P

paradoxist geometry 309

path 81

p-element 28

p-subgroup 28

permutation 16

permutation groups 42

permutation pair 286

permutation representation 42

planar graph 87

planar regular tessellation 199

point 118

polygonal presentation 128

power set 3

primitive extended 71

primitive group 61

projection 124, 212

projective plane 124

pseudo- Euclidean space 335

pure rotation 168

Q

quadricell 175

quotient group 23

quotient multigroup 34

quotient space 123




quotient topology 123

R

reflection 320, 334, 341

regular group 46

regular map 191

regular representation 19

Riemann manifold 358

Riemann surface 147

right action 42

root 268

rooted map 272

rooted map sequence 278

rooted orientable map polynomial 275

rooted non-orientable map polynomial 275

rooted total map polynomial 275

root polynomial 268

rotation 320, 334, 341

rotation embedding scheme 168

rotation matrix 336

rotation system 171

S

Schwarzschild-ADS spacetime 362

second countable 118

Seifert- Van Kampen theorem 141

set 2semi-arc automorphism 104

semi-arc set 103

semi-arc transitive 105

semigroup 7

semidirect product 25

semi-regular group 46

semi-regular graph 256

semi-regular map 256

sharply k-transitive group 47similarity 42

simply connected space 140

simple graph 80

simple group 15

signature 155

size 80

s-line 308, 336

Smarandache geometry 307Smarandache graphoidal cover 89

Smarandache graphoidal tree cover 89

Smarandache manifold 307

Smarandache multi-space 40

Smarandache plane 321

Smarandachely automorphism 317

Smarandachely chromatic number 89

Smarandachely coloring 89

Smarandachely decomposition 89

Smarandachely denied 307

Smarandachely embeddable graph 89

Smarandachely embedded graph 338

Smarandachely graph 89

Smarandachely map 325

Smarandachely path cover 89

Smarandachely property 89

socle 66

spanning subgraph 83

spherical tessellation 200

s-plane 308

s-space 308



Index 383

s-triangle 308

stabilizer 42

standard map 179

straight-line homotopy 137subgraph 82

subgroup 12

submultigroup 31

subset 3

surface 121

surface group 143

surface presentation 128

surjection 4Sylow p-subgroup 28

T

topology 118

total genus polynomial 276

Townsend-Wohlfarth type metric 360

trail 81

triangulation 126

transitive group 45

translation 320, 334, 340

transitive root 268

U

union set 2

vertex cover 86

vertex independent set 86

vertex set 80

voltage graph 215voltage map 216

W

walk 81

Documents

Automorphism Groups of Maps, Surfaces and Smarandache Geometries (second edition), graduate text book in mathematics, by Linfan Mao